Elliptic curve - Wikipedia

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cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle The group law subsection </button> <ul id="toc-The_group_law-sublist" class="vector-toc-list"> <li id="toc-Algebraic_interpretation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Algebraic_interpretation"> <div class="vector-toc-text"> 2.1 Algebraic interpretation </div> </a> <ul id="toc-Algebraic_interpretation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Non-Weierstrass_curves" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Non-Weierstrass_curves"> <div class="vector-toc-text"> 2.2 Non-Weierstrass curves </div> </a> <ul id="toc-Non-Weierstrass_curves-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Elliptic_curves_over_the_rational_numbers" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Elliptic_curves_over_the_rational_numbers"> <div class="vector-toc-text"> 3 Elliptic curves over the rational numbers </div> </a> <button aria-controls="toc-Elliptic_curves_over_the_rational_numbers-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Elliptic curves over the rational numbers subsection </button> <ul id="toc-Elliptic_curves_over_the_rational_numbers-sublist" class="vector-toc-list"> <li id="toc-Integral_points" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Integral_points"> <div class="vector-toc-text"> 3.1 Integral points </div> </a> <ul id="toc-Integral_points-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_structure_of_rational_points" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_structure_of_rational_points"> <div class="vector-toc-text"> 3.2 The structure of rational points </div> </a> <ul id="toc-The_structure_of_rational_points-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_Birch_and_Swinnerton-Dyer_conjecture" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_Birch_and_Swinnerton-Dyer_conjecture"> <div class="vector-toc-text"> 3.3 The Birch and Swinnerton-Dyer conjecture </div> </a> <ul id="toc-The_Birch_and_Swinnerton-Dyer_conjecture-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Elliptic_curves_over_finite_fields" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Elliptic_curves_over_finite_fields"> <div class="vector-toc-text"> 4 Elliptic curves over finite fields </div> </a> <ul id="toc-Elliptic_curves_over_finite_fields-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Elliptic_curves_over_a_general_field" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Elliptic_curves_over_a_general_field"> <div class="vector-toc-text"> 5 Elliptic curves over a general field </div> </a> <ul id="toc-Elliptic_curves_over_a_general_field-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Elliptic_curves_over_the_complex_numbers" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Elliptic_curves_over_the_complex_numbers"> <div class="vector-toc-text"> 6 Elliptic curves over the complex numbers </div> </a> <ul id="toc-Elliptic_curves_over_the_complex_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_dual_isogeny" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#The_dual_isogeny"> <div class="vector-toc-text"> 7 The dual isogeny </div> </a> <button aria-controls="toc-The_dual_isogeny-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle The dual isogeny subsection </button> <ul id="toc-The_dual_isogeny-sublist" class="vector-toc-list"> <li id="toc-Construction_of_the_dual_isogeny" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Construction_of_the_dual_isogeny"> <div class="vector-toc-text"> 7.1 Construction of the dual isogeny </div> </a> <ul id="toc-Construction_of_the_dual_isogeny-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Algorithms_that_use_elliptic_curves" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Algorithms_that_use_elliptic_curves"> <div class="vector-toc-text"> 8 Algorithms that use elliptic curves </div> </a> <ul id="toc-Algorithms_that_use_elliptic_curves-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Alternative_representations_of_elliptic_curves" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Alternative_representations_of_elliptic_curves"> <div class="vector-toc-text"> 9 Alternative representations of elliptic curves </div> </a> <ul id="toc-Alternative_representations_of_elliptic_curves-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 10 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 11 Notes </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 12 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 13 External links </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button 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Available in 36 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-36" aria-hidden="true" > 36 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D9%86%D8%AD%D9%86%D9%89_%D8%A5%D9%87%D9%84%D9%8A%D9%84%D8%AC%D9%8A" title="منحنى إهليلجي – Arabic" lang="ar" hreflang="ar" data-title="منحنى إهليلجي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%AD%D0%BB%D1%96%D0%BF%D1%82%D1%8B%D1%87%D0%BD%D0%B0%D1%8F_%D0%BA%D1%80%D1%8B%D0%B2%D0%B0%D1%8F" title="Эліптычная крывая – Belarusian" lang="be" hreflang="be" data-title="Эліптычная крывая" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Corba_el%C2%B7l%C3%ADptica" title="Corba el·líptica – Catalan" lang="ca" hreflang="ca" data-title="Corba el·líptica" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%AD%D0%BB%D0%BB%D0%B8%D0%BF%D1%81%D0%BB%D0%B0_%D0%B9%C4%95%D1%80" title="Эллипсла йĕр – Chuvash" lang="cv" hreflang="cv" data-title="Эллипсла йĕр" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target">Чӑвашла</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Eliptick%C3%A1_k%C5%99ivka" title="Eliptická křivka – Czech" lang="cs" hreflang="cs" data-title="Eliptická křivka" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Elliptisk_kurve" title="Elliptisk kurve – Danish" lang="da" hreflang="da" data-title="Elliptisk kurve" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Elliptische_Kurve" title="Elliptische Kurve – German" lang="de" hreflang="de" data-title="Elliptische Kurve" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Elliptiline_joon" title="Elliptiline joon – Estonian" lang="et" hreflang="et" data-title="Elliptiline joon" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%95%CE%BB%CE%BB%CE%B5%CE%B9%CF%80%CF%84%CE%B9%CE%BA%CE%AE_%CE%BA%CE%B1%CE%BC%CF%80%CF%8D%CE%BB%CE%B7" title="Ελλειπτική καμπύλη – Greek" lang="el" hreflang="el" data-title="Ελλειπτική καμπύλη" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Curva_el%C3%ADptica" title="Curva elíptica – Spanish" lang="es" hreflang="es" data-title="Curva elíptica" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Elipsa_kurbo" title="Elipsa kurbo – Esperanto" lang="eo" hreflang="eo" data-title="Elipsa kurbo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AE%D9%85_%D8%A8%DB%8C%D8%B6%D9%88%DB%8C" title="خم بیضوی – Persian" lang="fa" hreflang="fa" data-title="خم بیضوی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Courbe_elliptique" title="Courbe elliptique – French" lang="fr" hreflang="fr" data-title="Courbe elliptique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%83%80%EC%9B%90%EA%B3%A1%EC%84%A0" title="타원곡선 – Korean" lang="ko" hreflang="ko" data-title="타원곡선" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Kurva_eliptik" title="Kurva eliptik – Indonesian" lang="id" hreflang="id" data-title="Kurva eliptik" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Sporger_ferill" title="Sporger ferill – Icelandic" lang="is" hreflang="is" data-title="Sporger ferill" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target">Íslenska</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Curva_ellittica" title="Curva ellittica – Italian" lang="it" hreflang="it" data-title="Curva ellittica" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A2%D7%A7%D7%95%D7%9D_%D7%90%D7%9C%D7%99%D7%A4%D7%98%D7%99" title="עקום אליפטי – Hebrew" lang="he" hreflang="he" data-title="עקום אליפטי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Elliptikus_g%C3%B6rbe" title="Elliptikus görbe – Hungarian" lang="hu" hreflang="hu" data-title="Elliptikus görbe" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Elliptische_kromme" title="Elliptische kromme – Dutch" lang="nl" hreflang="nl" data-title="Elliptische kromme" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%A5%95%E5%86%86%E6%9B%B2%E7%B7%9A" title="楕円曲線 – Japanese" lang="ja" hreflang="ja" data-title="楕円曲線" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Elliptisk_kurve" title="Elliptisk kurve – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Elliptisk kurve" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Krzywa_eliptyczna" title="Krzywa eliptyczna – Polish" lang="pl" hreflang="pl" data-title="Krzywa eliptyczna" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Curva_el%C3%ADptica" title="Curva elíptica – Portuguese" lang="pt" hreflang="pt" data-title="Curva elíptica" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Curbe_eliptice" title="Curbe eliptice – Romanian" lang="ro" hreflang="ro" data-title="Curbe eliptice" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru badge-Q17559452 badge-recommendedarticle mw-list-item" title="recommended article"><a href="https://ru.wikipedia.org/wiki/%D0%AD%D0%BB%D0%BB%D0%B8%D0%BF%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D0%BA%D1%80%D0%B8%D0%B2%D0%B0%D1%8F" title="Эллиптическая кривая – Russian" lang="ru" hreflang="ru" data-title="Эллиптическая кривая" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Curva_ellittica" title="Curva ellittica – Sicilian" lang="scn" hreflang="scn" data-title="Curva ellittica" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target">Sicilianu</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Elliptic_curve" title="Elliptic curve – Simple English" lang="en-simple" hreflang="en-simple" data-title="Elliptic curve" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Elipti%C4%8Dna_krivulja" title="Eliptična krivulja – Slovenian" lang="sl" hreflang="sl" data-title="Eliptična krivulja" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%95%D0%BB%D0%B8%D0%BF%D1%82%D0%B8%D1%87%D0%BA%D0%B0_%D0%BA%D1%80%D0%B8%D0%B2%D0%B0" title="Елиптичка крива – Serbian" lang="sr" hreflang="sr" data-title="Елиптичка крива" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Elliptinen_k%C3%A4yr%C3%A4" title="Elliptinen käyrä – Finnish" lang="fi" hreflang="fi" data-title="Elliptinen käyrä" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Elliptisk_kurva" title="Elliptisk kurva – Swedish" lang="sv" hreflang="sv" data-title="Elliptisk kurva" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%80%E0%B8%AA%E0%B9%89%E0%B8%99%E0%B9%82%E0%B8%84%E0%B9%89%E0%B8%87%E0%B9%80%E0%B8%8A%E0%B8%B4%E0%B8%87%E0%B8%A7%E0%B8%87%E0%B8%A3%E0%B8%B5" title="เส้นโค้งเชิงวงรี – Thai" lang="th" hreflang="th" data-title="เส้นโค้งเชิงวงรี" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%95%D0%BB%D1%96%D0%BF%D1%82%D0%B8%D1%87%D0%BD%D0%B0_%D0%BA%D1%80%D0%B8%D0%B2%D0%B0" title="Еліптична крива – Ukrainian" lang="uk" hreflang="uk" data-title="Еліптична крива" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90%C6%B0%E1%BB%9Dng_cong_elliptic" title="Đường cong elliptic – Vietnamese" lang="vi" hreflang="vi" data-title="Đường cong elliptic" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%A4%AD%E5%9C%86%E6%9B%B2%E7%BA%BF" title="椭圆曲线 – Chinese" lang="zh" hreflang="zh" data-title="椭圆曲线" data-language-autonym="中文" data-language-local-name="Chinese" 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div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Ellipse" title="Ellipse">Ellipse</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">"Elliptic Equation" redirects here. For the type of partial differential equation, see <a href="/wiki/Elliptic_partial_differential_equation" title="Elliptic partial differential equation">Elliptic partial differential equation</a>.</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:EllipticCurveCatalog.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/db/EllipticCurveCatalog.svg/392px-EllipticCurveCatalog.svg.png" decoding="async" width="392" height="379" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/db/EllipticCurveCatalog.svg/588px-EllipticCurveCatalog.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/db/EllipticCurveCatalog.svg/784px-EllipticCurveCatalog.svg.png 2x" data-file-width="533" data-file-height="515" /></a><figcaption>A catalog of elliptic curves. The region shown is x, y ∈ [−3,3]. 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theory</a></th></tr><tr><td class="sidebar-image"><a href="/wiki/File:Cyclic_group.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Cyclic_group.svg/120px-Cyclic_group.svg.png" decoding="async" width="120" height="117" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Cyclic_group.svg/180px-Cyclic_group.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Cyclic_group.svg/240px-Cyclic_group.svg.png 2x" data-file-width="443" data-file-height="431" /></a></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)">Basic notions</div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base, #202122 ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Subgroup" title="Subgroup">Subgroup</a></li> <li><a href="/wiki/Normal_subgroup" title="Normal subgroup">Normal subgroup</a></li> <li><a href="/wiki/Group_action" title="Group action">Group action</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Quotient_group" title="Quotient group">Quotient group</a></li> <li><a href="/wiki/Semidirect_product" title="Semidirect product">(Semi-)</a><a href="/wiki/Direct_product_of_groups" title="Direct product of groups">direct product</a></li> <li><a href="/wiki/Direct_sum_of_groups" title="Direct sum of groups">Direct sum</a></li> <li><a href="/wiki/Free_product" title="Free product">Free product</a></li> <li><a href="/wiki/Wreath_product" title="Wreath product">Wreath product</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Group_homomorphism" title="Group homomorphism">Group homomorphisms</a></th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Kernel_(algebra)#Group_homomorphisms" title="Kernel (algebra)">kernel</a></li> <li><a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Simple_group" title="Simple group">simple</a></li> <li><a href="/wiki/Finite_group" title="Finite group">finite</a></li> <li><a href="/wiki/Infinite_group" title="Infinite group">infinite</a></li> <li><a href="/wiki/Continuous_group" class="mw-redirect" title="Continuous group">continuous</a></li> <li><a href="/wiki/Multiplicative_group" title="Multiplicative group">multiplicative</a></li> <li><a href="/wiki/Additive_group" title="Additive group">additive</a></li> <li><a href="/wiki/Cyclic_group" title="Cyclic group">cyclic</a></li> <li><a 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style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base, #202122 ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Cyclic_group" title="Cyclic group">Cyclic group</a> Zn</li> <li><a href="/wiki/Symmetric_group" title="Symmetric group">Symmetric group</a> Sn</li> <li><a href="/wiki/Alternating_group" title="Alternating group">Alternating group</a> An</li></ul> <ul><li><a href="/wiki/Dihedral_group" title="Dihedral group">Dihedral group</a> Dn</li> <li><a href="/wiki/Quaternion_group" title="Quaternion group">Quaternion group</a> Q</li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Cauchy%27s_theorem_(group_theory)" title="Cauchy's theorem (group theory)">Cauchy's theorem</a></li> <li><a href="/wiki/Lagrange%27s_theorem_(group_theory)" title="Lagrange's theorem (group theory)">Lagrange's theorem</a></li></ul> <ul><li><a href="/wiki/Sylow_theorems" title="Sylow theorems">Sylow theorems</a></li> <li><a href="/wiki/Hall_subgroup" title="Hall subgroup">Hall's theorem</a></li></ul> <ul><li><a href="/wiki/P-group" title="P-group">p-group</a></li> <li><a href="/wiki/Elementary_abelian_group" title="Elementary abelian group">Elementary abelian group</a></li></ul> <ul><li><a href="/wiki/Frobenius_group" title="Frobenius group">Frobenius group</a></li></ul> <ul><li><a href="/wiki/Schur_multiplier" title="Schur multiplier">Schur multiplier</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Classification_of_finite_simple_groups" title="Classification of finite simple groups">Classification of finite simple groups</a></th></tr><tr><td class="sidebar-content"> <ul><li>cyclic</li> <li>alternating</li> <li><a href="/wiki/Group_of_Lie_type" title="Group of Lie type">Lie type</a></li> <li><a href="/wiki/Sporadic_group" title="Sporadic group">sporadic</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><div class="hlist"><ul><li><a href="/wiki/Discrete_group" title="Discrete group">Discrete groups</a></li><li><a href="/wiki/Lattice_(discrete_subgroup)" title="Lattice (discrete subgroup)">Lattices</a></li></ul></div></div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Integer" title="Integer">Integers</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" />)</li> <li><a href="/wiki/Free_group" title="Free group">Free group</a></li></ul> <div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><a href="/wiki/Modular_group" title="Modular group">Modular groups</a> <div class="hlist"><ul><li>PSL(2, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" />)</li><li>SL(2, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" />)</li></ul></div></div> <ul><li><a href="/wiki/Arithmetic_group" title="Arithmetic group">Arithmetic group</a></li> <li><a href="/wiki/Lattice_(group)" title="Lattice (group)">Lattice</a></li> <li><a href="/wiki/Hyperbolic_group" title="Hyperbolic group">Hyperbolic group</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Topological_group" title="Topological group">Topological</a> and <a href="/wiki/Lie_group" title="Lie group">Lie groups</a></div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Solenoid_(mathematics)" title="Solenoid (mathematics)">Solenoid</a></li> <li><a href="/wiki/Circle_group" title="Circle group">Circle</a></li></ul> <ul><li><a href="/wiki/General_linear_group" title="General linear group">General linear</a> GL(n)</li></ul> <ul><li><a href="/wiki/Special_linear_group" title="Special linear group">Special linear</a> SL(n)</li></ul> <ul><li><a href="/wiki/Orthogonal_group" title="Orthogonal group">Orthogonal</a> O(n)</li></ul> <ul><li><a href="/wiki/Euclidean_group" title="Euclidean group">Euclidean</a> E(n)</li></ul> <ul><li><a href="/wiki/Special_orthogonal_group" class="mw-redirect" title="Special orthogonal group">Special orthogonal</a> SO(n)</li></ul> <ul><li><a href="/wiki/Unitary_group" title="Unitary group">Unitary</a> U(n)</li></ul> <ul><li><a href="/wiki/Special_unitary_group" title="Special unitary group">Special unitary</a> SU(n)</li></ul> <ul><li><a href="/wiki/Symplectic_group" title="Symplectic group">Symplectic</a> Sp(n)</li></ul> <ul><li><a href="/wiki/G2_(mathematics)" title="G2 (mathematics)">G2</a></li> <li><a href="/wiki/F4_(mathematics)" title="F4 (mathematics)">F4</a></li> <li><a href="/wiki/E6_(mathematics)" title="E6 (mathematics)">E6</a></li> <li><a href="/wiki/E7_(mathematics)" title="E7 (mathematics)">E7</a></li> <li><a href="/wiki/E8_(mathematics)" title="E8 (mathematics)">E8</a></li></ul> <ul><li><a href="/wiki/Lorentz_group" title="Lorentz group">Lorentz</a></li> <li><a href="/wiki/Poincar%C3%A9_group" title="Poincaré group">Poincaré</a></li> <li><a href="/wiki/Conformal_group" title="Conformal group">Conformal</a></li></ul> <ul><li><a href="/wiki/Diffeomorphism" title="Diffeomorphism">Diffeomorphism</a></li> <li><a href="/wiki/Loop_group" title="Loop group">Loop</a></li></ul> <div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><a href="/wiki/Infinite_dimensional_Lie_group" class="mw-redirect" title="Infinite dimensional Lie group">Infinite dimensional Lie group</a> <div class="hlist"><ul><li>O(∞)</li><li>SU(∞)</li><li>Sp(∞)</li></ul></div></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Algebraic_group" title="Algebraic group">Algebraic groups</a></div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Linear_algebraic_group" title="Linear algebraic group">Linear algebraic group</a></li></ul> <ul><li><a href="/wiki/Reductive_group" title="Reductive group">Reductive group</a></li></ul> <ul><li><a href="/wiki/Abelian_variety" title="Abelian variety">Abelian variety</a></li></ul> <ul><li><a class="mw-selflink selflink">Elliptic curve</a></li></ul></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Group_theory_sidebar" title="Template:Group theory sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Group_theory_sidebar" title="Template talk:Group theory sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Group_theory_sidebar" title="Special:EditPage/Template:Group theory sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, an elliptic curve is a <a href="/wiki/Smoothness" title="Smoothness">smooth</a>, <a href="/wiki/Projective_variety" title="Projective variety">projective</a>, <a href="/wiki/Algebraic_curve" title="Algebraic curve">algebraic curve</a> of <a href="/wiki/Genus_of_an_algebraic_curve" class="mw-redirect" title="Genus of an algebraic curve">genus</a> one, on which there is a specified point O. An elliptic curve is defined over a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> K and describes points in K2, the <a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a> of K with itself. If the field's <a href="/wiki/Characteristic_(algebra)" title="Characteristic (algebra)">characteristic</a> is different from 2 and 3, then the curve can be described as a <a href="/wiki/Plane_algebraic_curve" class="mw-redirect" title="Plane algebraic curve">plane algebraic curve</a> which consists of solutions (x, y) for: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y^{2}=x^{3}+ax+b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mi>a</mi> <mi>x</mi> <mo>+</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{2}=x^{3}+ax+b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dbe6cab1bc2c7f1c99757dc6e5d7a517cf9b4f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.935ex; height:3.009ex;" alt="{\displaystyle y^{2}=x^{3}+ax+b}" /></dd></dl> for some coefficients a and b in K. The curve is required to be <a href="/wiki/Singular_point_of_a_curve" title="Singular point of a curve">non-singular</a>, which means that the curve has no <a href="/wiki/Cusp_(singularity)" title="Cusp (singularity)">cusps</a> or <a href="/wiki/Self-intersection" class="mw-redirect" title="Self-intersection">self-intersections</a>. (This is equivalent to the condition 4a3 + 27b2 ≠ 0, that is, being <a href="/wiki/Square-free_polynomial" title="Square-free polynomial">square-free</a> in x.) It is always understood that the curve is really sitting in the <a href="/wiki/Projective_plane" title="Projective plane">projective plane</a>, with the point O being the unique <a href="/wiki/Point_at_infinity" title="Point at infinity">point at infinity</a>. Many sources define an elliptic curve to be simply a curve given by an equation of this form. (When the <a href="/wiki/Coefficient_field" class="mw-redirect" title="Coefficient field">coefficient field</a> has characteristic 2 or 3, the above equation is not quite general enough to include all non-singular <a href="/wiki/Cubic_plane_curve" title="Cubic plane curve">cubic curves</a>; see <a href="#Elliptic_curves_over_a_general_field">§ Elliptic curves over a general field</a> below.) An elliptic curve is an <a href="/wiki/Abelian_variety" title="Abelian variety">abelian variety</a> – that is, it has a group law defined algebraically, with respect to which it is an <a href="/wiki/Abelian_group" title="Abelian group">abelian group</a> – and O serves as the identity element. If y2 = P(x), where P is any polynomial of degree three in x with no repeated roots, the solution set is a nonsingular plane curve of <a href="/wiki/Genus_(mathematics)" title="Genus (mathematics)">genus</a> one, an elliptic curve. If P has degree four and is <a href="/wiki/Square-free_polynomial" title="Square-free polynomial">square-free</a> this equation again describes a plane curve of genus one; however, it has no natural choice of identity element. More generally, any algebraic curve of genus one, for example the intersection of two <a href="/wiki/Quadric_(algebraic_geometry)" title="Quadric (algebraic geometry)">quadric surfaces</a> embedded in three-dimensional projective space, is called an elliptic curve, provided that it is equipped with a marked point to act as the identity. Using the theory of <a href="/wiki/Elliptic_function" title="Elliptic function">elliptic functions</a>, it can be shown that elliptic curves defined over the <a href="/wiki/Complex_number" title="Complex number">complex numbers</a> correspond to embeddings of the <a href="/wiki/Torus" title="Torus">torus</a> into the <a href="/wiki/Complex_projective_plane" title="Complex projective plane">complex projective plane</a>. The torus is also an <a href="/wiki/Abelian_group" title="Abelian group">abelian group</a>, and this correspondence is also a <a href="/wiki/Group_isomorphism" title="Group isomorphism">group isomorphism</a>. Elliptic curves are especially important in <a href="/wiki/Number_theory" title="Number theory">number theory</a>, and constitute a major area of current research; for example, they were used in <a href="/wiki/Wiles%27s_proof_of_Fermat%27s_Last_Theorem" title="Wiles's proof of Fermat's Last Theorem">Andrew Wiles's proof of Fermat's Last Theorem</a>. They also find applications in <a href="/wiki/Elliptic_curve_cryptography" class="mw-redirect" title="Elliptic curve cryptography">elliptic curve cryptography</a> (ECC) and <a href="/wiki/Integer_factorization" title="Integer factorization">integer factorization</a>. An elliptic curve is not an <a href="/wiki/Ellipse" title="Ellipse">ellipse</a> in the sense of a projective conic, which has genus zero: see <a href="/wiki/Elliptic_integral" title="Elliptic integral">elliptic integral</a> for the origin of the term. However, there is a natural representation of real elliptic curves with shape invariant j ≥ 1 as ellipses in the hyperbolic plane <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {H} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {H} ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd949bba96edbb143fd05b712e274aa07ab1c75f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.862ex; height:2.676ex;" alt="{\displaystyle \mathbb {H} ^{2}}" />. Specifically, the intersections of the Minkowski hyperboloid with quadric surfaces characterized by a certain constant-angle property produce the Steiner ellipses in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {H} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {H} ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd949bba96edbb143fd05b712e274aa07ab1c75f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.862ex; height:2.676ex;" alt="{\displaystyle \mathbb {H} ^{2}}" /> (generated by orientation-preserving collineations). Further, the orthogonal trajectories of these ellipses comprise the elliptic curves with j ≤ 1, and any ellipse in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {H} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {H} ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd949bba96edbb143fd05b712e274aa07ab1c75f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.862ex; height:2.676ex;" alt="{\displaystyle \mathbb {H} ^{2}}" /> described as a locus relative to two foci is uniquely the elliptic curve sum of two Steiner ellipses, obtained by adding the pairs of intersections on each orthogonal trajectory. Here, the vertex of the hyperboloid serves as the identity on each trajectory curve.<a href="#cite_note-1">[1]</a> <a href="/wiki/Topologically" class="mw-redirect" title="Topologically">Topologically</a>, a complex elliptic curve is a <a href="/wiki/Torus" title="Torus">torus</a>, while a complex ellipse is a <a href="/wiki/Sphere" title="Sphere">sphere</a>. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Elliptic_curves_over_the_real_numbers">Elliptic curves over the real numbers</h2>[<a href="/w/index.php?title=Elliptic_curve&action=edit&section=1" title="Edit section: Elliptic curves over the real numbers">edit</a>]</div> <figure class="mw-halign-right" typeof="mw:File/Frame"><a href="/wiki/File:ECClines-3.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d0/ECClines-3.svg/335px-ECClines-3.svg.png" decoding="async" width="335" height="190" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d0/ECClines-3.svg/503px-ECClines-3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d0/ECClines-3.svg/670px-ECClines-3.svg.png 2x" data-file-width="335" data-file-height="190" /></a><figcaption>Graphs of curves y2 = x3 − x and y2 = x3 − x + 1</figcaption></figure> Although the formal definition of an elliptic curve requires some background in <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, it is possible to describe some features of elliptic curves over the <a href="/wiki/Real_number" title="Real number">real numbers</a> using only introductory <a href="/wiki/Algebra" title="Algebra">algebra</a> and <a href="/wiki/Geometry" title="Geometry">geometry</a>. In this context, an elliptic curve is a <a href="/wiki/Plane_curve" title="Plane curve">plane curve</a> defined by an equation of the form <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y^{2}=x^{3}+ax+b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mi>a</mi> <mi>x</mi> <mo>+</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{2}=x^{3}+ax+b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dbe6cab1bc2c7f1c99757dc6e5d7a517cf9b4f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.935ex; height:3.009ex;" alt="{\displaystyle y^{2}=x^{3}+ax+b}" /></dd></dl> after a linear change of variables (a and b are real numbers). This type of equation is called a Weierstrass equation, and said to be in Weierstrass form, or Weierstrass normal form. The definition of elliptic curve also requires that the curve be <a href="/wiki/Singular_point_of_an_algebraic_variety" title="Singular point of an algebraic variety">non-singular</a>. Geometrically, this means that the graph has no <a href="/wiki/Cusp_(singularity)" title="Cusp (singularity)">cusps</a>, self-intersections, or <a href="/wiki/Isolated_point" title="Isolated point">isolated points</a>. Algebraically, this holds if and only if the <a href="/wiki/Discriminant" title="Discriminant">discriminant</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32769037c408874e1890f77554c65f39c523ebe2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }" />, is not equal to zero. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta =-16\left(4a^{3}+27b^{2}\right)\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>=</mo> <mo>−</mo> <mn>16</mn> <mrow> <mo>(</mo> <mrow> <mn>4</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>27</mn> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta =-16\left(4a^{3}+27b^{2}\right)\neq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7134561ef74fb54f73f9ce7c77811bdf5f3e3c8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.609ex; height:3.343ex;" alt="{\displaystyle \Delta =-16\left(4a^{3}+27b^{2}\right)\neq 0}" /></dd></dl> The discriminant is zero when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=-3k^{2},b=2k^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mo>−</mo> <mn>3</mn> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mi>b</mi> <mo>=</mo> <mn>2</mn> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=-3k^{2},b=2k^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acafac5026e6175f5366c8c498d64955788a1f6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.122ex; height:3.009ex;" alt="{\displaystyle a=-3k^{2},b=2k^{3}}" />. (Although the factor −16 is irrelevant to whether or not the curve is non-singular, this definition of the discriminant is useful in a more advanced study of elliptic curves.)<a href="#cite_note-2">[2]</a> The real graph of a non-singular curve has two components if its discriminant is positive, and one component if it is negative. For example, in the graphs shown in figure to the right, the discriminant in the first case is 64, and in the second case is −368. Following the convention at <a href="/wiki/Conic_section#Discriminant" title="Conic section">Conic_section#Discriminant</a>, elliptic curves require that the discriminant is negative. <div class="mw-heading mw-heading2"><h2 id="The_group_law">The group law</h2>[<a href="/w/index.php?title=Elliptic_curve&action=edit&section=2" title="Edit section: The group law">edit</a>]</div> When working in the <a href="/wiki/Projective_plane" title="Projective plane">projective plane</a>, the equation in <a href="/wiki/Homogeneous_coordinates" title="Homogeneous coordinates">homogeneous coordinates</a> becomes <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {Y^{2}}{Z^{2}}}={\frac {X^{3}}{Z^{3}}}+a{\frac {X}{Z}}+b.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>X</mi> <mi>Z</mi> </mfrac> </mrow> <mo>+</mo> <mi>b</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {Y^{2}}{Z^{2}}}={\frac {X^{3}}{Z^{3}}}+a{\frac {X}{Z}}+b.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1014b376ba8c7b2a83b6bfce12f9d9f16c392d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:22.147ex; height:5.843ex;" alt="{\displaystyle {\frac {Y^{2}}{Z^{2}}}={\frac {X^{3}}{Z^{3}}}+a{\frac {X}{Z}}+b.}" /></dd></dl> This equation is not defined on the <a href="/wiki/Line_at_infinity" title="Line at infinity">line at infinity</a>, but we can multiply by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/456af6dd074780aadcc2ef296fa0c1dbe523fd75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.763ex; height:2.676ex;" alt="{\displaystyle Z^{3}}" /> to get one that is: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ZY^{2}=X^{3}+aZ^{2}X+bZ^{3}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <msup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mi>a</mi> <msup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>X</mi> <mo>+</mo> <mi>b</mi> <msup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ZY^{2}=X^{3}+aZ^{2}X+bZ^{3}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e03918ee5412ec1ab69b6e0c9cf22dd7b8ba92f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:26.845ex; height:2.843ex;" alt="{\displaystyle ZY^{2}=X^{3}+aZ^{2}X+bZ^{3}.}" /></dd></dl> This resulting equation is defined on the whole projective plane, and the curve it defines projects onto the elliptic curve of interest. To find its intersection with the line at infinity, we can just posit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49175243147048b9854d4f1e640e3e0da82c09ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.941ex; height:2.176ex;" alt="{\displaystyle Z=0}" />. This implies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{3}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{3}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ef14a2e2ef0eb207bda55c66ead31bda29cb120" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.312ex; height:2.676ex;" alt="{\displaystyle X^{3}=0}" />, which in a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> means <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d519e9e94f279ea82581dfa70a2444e896e2d860" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.241ex; height:2.176ex;" alt="{\displaystyle X=0}" />. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" /> on the other hand can take any value, and thus all triplets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,Y,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>Y</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,Y,0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2ebd0e1198b117de82d6c953cdc78b0cca855e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.975ex; height:2.843ex;" alt="{\displaystyle (0,Y,0)}" /> satisfy the equation. In projective geometry this set is simply the point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O=[0:1:0]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>:</mo> <mn>1</mn> <mo>:</mo> <mn>0</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O=[0:1:0]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18273b8a82c7523639ad09fbbd45bd749d2fdb14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.527ex; height:2.843ex;" alt="{\displaystyle O=[0:1:0]}" />, which is thus the unique intersection of the curve with the line at infinity. Since the curve is smooth, hence <a href="/wiki/Continuous_function" title="Continuous function">continuous</a>, it can be shown that this point at infinity is the identity element of a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> structure whose operation is geometrically described as follows: Since the curve is symmetric about the x axis, given any point P, we can take −P to be the point opposite it. We then have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -O=O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>O</mi> <mo>=</mo> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -O=O}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cdda9c82d463006b0da1f490e2322bd1004743f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.453ex; height:2.343ex;" alt="{\displaystyle -O=O}" />, as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="{\displaystyle O}" /> lies on the XZ plane, so that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -O}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df10fb57bce3d9e2e6a9264bb1b73f13683abb10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.581ex; height:2.343ex;" alt="{\displaystyle -O}" /> is also the symmetrical of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="{\displaystyle O}" /> about the origin, and thus represents the same projective point. If P and Q are two points on the curve, then we can uniquely describe a third point P + Q in the following way. First, draw the line that intersects P and Q. This will generally intersect the cubic at a third point, R. We then take P + Q to be −R, the point opposite R. This definition for addition works except in a few special cases related to the point at infinity and intersection multiplicity. The first is when one of the points is O. Here, we define P + O = P = O + P, making O the identity of the group. If P = Q, we only have one point, thus we cannot define the line between them. In this case, we use the tangent line to the curve at this point as our line. In most cases, the tangent will intersect a second point R, and we can take its opposite. If P and Q are opposites of each other, we define P + Q = O. Lastly, if P is an <a href="/wiki/Inflection_point" title="Inflection point">inflection point</a> (a point where the concavity of the curve changes), we take R to be P itself, and P + P is simply the point opposite itself, i.e. itself. <figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:ECClines.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c1/ECClines.svg/680px-ECClines.svg.png" decoding="async" width="680" height="206" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c1/ECClines.svg/1020px-ECClines.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c1/ECClines.svg/1360px-ECClines.svg.png 2x" data-file-width="680" data-file-height="206" /></a><figcaption></figcaption></figure> Let K be a field over which the curve is defined (that is, the coefficients of the defining equation or equations of the curve are in K) and denote the curve by E. Then the K-<a href="/wiki/Rational_point" title="Rational point">rational points</a> of E are the points on E whose coordinates all lie in K, including the point at infinity. The set of K-rational points is denoted by E(K). E(K) is a group, because properties of polynomial equations show that if P is in E(K), then −P is also in E(K), and if two of P, Q, R are in E(K), then so is the third. Additionally, if K is a subfield of L, then E(K) is a <a href="/wiki/Subgroup" title="Subgroup">subgroup</a> of E(L). <div class="mw-heading mw-heading3"><h3 id="Algebraic_interpretation">Algebraic interpretation</h3>[<a href="/w/index.php?title=Elliptic_curve&action=edit&section=3" title="Edit section: Algebraic interpretation">edit</a>]</div> The above groups can be described algebraically as well as geometrically. Given the curve y2 = x3 + bx + c over the field K (whose <a href="/wiki/Prime_subfield" class="mw-redirect" title="Prime subfield">characteristic</a> we assume to be neither 2 nor 3), and points P = (xP, yP) and Q = (xQ, yQ) on the curve, assume first that xP ≠ xQ (case 1). Let y = sx + d be the equation of the line that intersects P and Q, which has the following slope: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s={\frac {y_{P}-y_{Q}}{x_{P}-x_{Q}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> </mrow> </msub> </mrow> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s={\frac {y_{P}-y_{Q}}{x_{P}-x_{Q}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0aa5e425104f3901729a4d2245cf5a3f03b8850b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.17ex; height:6.009ex;" alt="{\displaystyle s={\frac {y_{P}-y_{Q}}{x_{P}-x_{Q}}}.}" /></dd></dl> The line equation and the curve equation intersect at the points xP, xQ, and xR, so the equations have identical y values at these values. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (sx+d)^{2}=x^{3}+bx+c,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>s</mi> <mi>x</mi> <mo>+</mo> <mi>d</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <mi>x</mi> <mo>+</mo> <mi>c</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (sx+d)^{2}=x^{3}+bx+c,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/661c8061ca819aa0445b0f19fad217f612f13762" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.484ex; height:3.176ex;" alt="{\displaystyle (sx+d)^{2}=x^{3}+bx+c,}" /></dd></dl> which is equivalent to <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{3}-s^{2}x^{2}-2sdx+bx+c-d^{2}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>s</mi> <mi>d</mi> <mi>x</mi> <mo>+</mo> <mi>b</mi> <mi>x</mi> <mo>+</mo> <mi>c</mi> <mo>−</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{3}-s^{2}x^{2}-2sdx+bx+c-d^{2}=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4314b4a280afc4343519e915b4b0fd8cef414bdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:36.427ex; height:2.843ex;" alt="{\displaystyle x^{3}-s^{2}x^{2}-2sdx+bx+c-d^{2}=0.}" /></dd></dl> Since xP, xQ, and xR are solutions, this equation has its roots at exactly the same x values as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x-x_{P})(x-x_{Q})(x-x_{R})=x^{3}+(-x_{P}-x_{Q}-x_{R})x^{2}+(x_{P}x_{Q}+x_{P}x_{R}+x_{Q}x_{R})x-x_{P}x_{Q}x_{R},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x-x_{P})(x-x_{Q})(x-x_{R})=x^{3}+(-x_{P}-x_{Q}-x_{R})x^{2}+(x_{P}x_{Q}+x_{P}x_{R}+x_{Q}x_{R})x-x_{P}x_{Q}x_{R},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82aa77860fd2c3904d6bad23883ee8447f5065de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:95.427ex; height:3.343ex;" alt="{\displaystyle (x-x_{P})(x-x_{Q})(x-x_{R})=x^{3}+(-x_{P}-x_{Q}-x_{R})x^{2}+(x_{P}x_{Q}+x_{P}x_{R}+x_{Q}x_{R})x-x_{P}x_{Q}x_{R},}" /></dd></dl> and because both equations are cubics, they must be the same polynomial up to a scalar. Then <a href="/wiki/Equating_the_coefficients" class="mw-redirect" title="Equating the coefficients">equating the coefficients</a> of x2 in both equations <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -s^{2}=(-x_{P}-x_{Q}-x_{R})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -s^{2}=(-x_{P}-x_{Q}-x_{R})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3a27705533351f294588c7c002fd6591df9e6a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.817ex; height:3.343ex;" alt="{\displaystyle -s^{2}=(-x_{P}-x_{Q}-x_{R})}" /></dd></dl> and solving for the unknown xR, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{R}=s^{2}-x_{P}-x_{Q}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{R}=s^{2}-x_{P}-x_{Q}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/391031c84855954bc232b3cac03dca0fcbf147c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.038ex; height:3.343ex;" alt="{\displaystyle x_{R}=s^{2}-x_{P}-x_{Q}.}" /></dd></dl> yR follows from the line equation <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{R}=y_{P}-s(x_{P}-x_{R}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>−</mo> <mi>s</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{R}=y_{P}-s(x_{P}-x_{R}),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4455b4037ab6b7ea322b154d26ca7a078aad7e2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.156ex; height:2.843ex;" alt="{\displaystyle y_{R}=y_{P}-s(x_{P}-x_{R}),}" /></dd></dl> and this is an element of K, because s is. If xP = xQ, then there are two options: if yP = −yQ (case 3), including the case where yP = yQ = 0 (case 4), then the sum is defined as 0; thus, the inverse of each point on the curve is found by reflecting it across the x axis. If yP = yQ ≠ 0, then Q = P and R = (xR, yR) = −(P + P) = −2P = −2Q (case 2 using P as R). The slope is given by the tangent to the curve at (xP, yP). <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}s&={\frac {3{x_{P}}^{2}+b}{2y_{P}}},\\x_{R}&=s^{2}-2x_{P},\\y_{R}&=y_{P}-s(x_{P}-x_{R}).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>s</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> </mrow> <mrow> <mn>2</mn> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>−</mo> <mi>s</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}s&={\frac {3{x_{P}}^{2}+b}{2y_{P}}},\\x_{R}&=s^{2}-2x_{P},\\y_{R}&=y_{P}-s(x_{P}-x_{R}).\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a2f7ec9946e7c84c08a5c75712a41d0bfa50479" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:24.098ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}s&={\frac {3{x_{P}}^{2}+b}{2y_{P}}},\\x_{R}&=s^{2}-2x_{P},\\y_{R}&=y_{P}-s(x_{P}-x_{R}).\end{aligned}}}" /></dd></dl> A more general expression for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" /> that works in both case 1 and case 2 is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s={\frac {{x_{P}}^{2}+x_{P}x_{Q}+{x_{Q}}^{2}+b}{y_{P}+y_{Q}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> </mrow> </msub> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> </mrow> <mrow> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s={\frac {{x_{P}}^{2}+x_{P}x_{Q}+{x_{Q}}^{2}+b}{y_{P}+y_{Q}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f40dca5c3f1fff98fb38d520bda863c16d78fb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.615ex; height:6.509ex;" alt="{\displaystyle s={\frac {{x_{P}}^{2}+x_{P}x_{Q}+{x_{Q}}^{2}+b}{y_{P}+y_{Q}}},}" /></dd></dl> where equality to <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style>⁠yP − yQ/xP − xQ⁠ relies on P and Q obeying y2 = x3 + bx + c. <div class="mw-heading mw-heading3"><h3 id="Non-Weierstrass_curves">Non-Weierstrass curves</h3>[<a href="/w/index.php?title=Elliptic_curve&action=edit&section=4" title="Edit section: Non-Weierstrass curves">edit</a>]</div> For the curve y2 = x3 + ax2 + bx + c (the general form of an elliptic curve with <a href="/wiki/Prime_subfield" class="mw-redirect" title="Prime subfield">characteristic</a> 3), the formulas are similar, with s = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" />⁠xP2 + xP xQ + xQ2 + axP + axQ + b/yP + yQ⁠ and xR = s2 − a − xP − xQ. For a general cubic curve not in Weierstrass normal form, we can still define a group structure by designating one of its nine inflection points as the identity O. In the projective plane, each line will intersect a cubic at three points when accounting for multiplicity. For a point P, −P is defined as the unique third point on the line passing through O and P. Then, for any P and Q, P + Q is defined as −R where R is the unique third point on the line containing P and Q. For an example of the group law over a non-Weierstrass curve, see <a href="/wiki/Hessian_form_of_an_elliptic_curve#Group_law" title="Hessian form of an elliptic curve">Hessian curves</a>. <div class="mw-heading mw-heading2"><h2 id="Elliptic_curves_over_the_rational_numbers">Elliptic curves over the rational numbers</h2>[<a href="/w/index.php?title=Elliptic_curve&action=edit&section=5" title="Edit section: Elliptic curves over the rational numbers">edit</a>]</div> A curve E defined over the field of rational numbers is also defined over the field of real numbers. Therefore, the law of addition (of points with real coordinates) by the tangent and secant method can be applied to E. The explicit formulae show that the sum of two points P and Q with rational coordinates has again rational coordinates, since the line joining P and Q has rational coefficients. This way, one shows that the set of rational points of E forms a subgroup of the group of real points of E. <div class="mw-heading mw-heading3"><h3 id="Integral_points">Integral points</h3>[<a href="/w/index.php?title=Elliptic_curve&action=edit&section=6" title="Edit section: Integral points">edit</a>]</div> This section is concerned with points P = (x, y) of E such that x is an integer. For example, the equation y2 = x3 + 17 has eight integral solutions with y > 0:<a href="#cite_note-3">[3]</a><a href="#cite_note-4">[4]</a> <dl><dd>(x, y) = (−2, 3), (−1, 4), (2, 5), (4, 9), (8, 23), (43, 282), (52, 375), (5234, 378661).</dd></dl> As another example, <a href="/wiki/Stella_octangula_number" title="Stella octangula number">Ljunggren's equation</a>, a curve whose Weierstrass form is y2 = x3 − 2x, has only four solutions with y ≥ 0 :<a href="#cite_note-5">[5]</a> <dl><dd>(x, y) = (0, 0), (−1, 1), (2, 2), (338, 6214).</dd></dl> <div class="mw-heading mw-heading3"><h3 id="The_structure_of_rational_points">The structure of rational points</h3>[<a href="/w/index.php?title=Elliptic_curve&action=edit&section=7" title="Edit section: The structure of rational points">edit</a>]</div> Rational points can be constructed by the method of tangents and secants detailed <a href="#The_group_law">above</a>, starting with a finite number of rational points. More precisely<a href="#cite_note-6">[6]</a> the <a href="/wiki/Mordell%E2%80%93Weil_theorem" title="Mordell–Weil theorem">Mordell–Weil theorem</a> states that the group E(Q) is a <a href="/wiki/Finitely_generated_group" title="Finitely generated group">finitely generated</a> (abelian) group. By the <a href="/wiki/Fundamental_theorem_of_finitely_generated_abelian_groups" class="mw-redirect" title="Fundamental theorem of finitely generated abelian groups">fundamental theorem of finitely generated abelian groups</a> it is therefore a finite direct sum of copies of Z and finite cyclic groups. The proof of the theorem<a href="#cite_note-7">[7]</a> involves two parts. The first part shows that for any integer m > 1, the <a href="/wiki/Quotient_group" title="Quotient group">quotient group</a> E(Q)/mE(Q) is finite (this is the weak Mordell–Weil theorem). Second, introducing a <a href="/wiki/Height_function" title="Height function">height function</a> h on the rational points E(Q) defined by h(P0) = 0 and h(P) = log max(|p|, |q|) if P (unequal to the point at infinity P0) has as <a href="/wiki/Abscissa" class="mw-redirect" title="Abscissa">abscissa</a> the rational number x = p/q (with <a href="/wiki/Coprime" class="mw-redirect" title="Coprime">coprime</a> p and q). This height function h has the property that h(mP) grows roughly like the square of m. Moreover, only finitely many rational points with height smaller than any constant exist on E. The proof of the theorem is thus a variant of the method of <a href="/wiki/Infinite_descent" class="mw-redirect" title="Infinite descent">infinite descent</a><a href="#cite_note-8">[8]</a> and relies on the repeated application of <a href="/wiki/Euclidean_algorithm" title="Euclidean algorithm">Euclidean divisions</a> on E: let P ∈ E(Q) be a rational point on the curve, writing P as the sum 2P1 + Q1 where Q1 is a fixed representant of P in E(Q)/2E(Q), the height of P1 is about <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" />⁠1/4⁠ of the one of P (more generally, replacing 2 by any m > 1, and <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" />⁠1/4⁠ by <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" />⁠1/m2⁠). Redoing the same with P1, that is to say P1 = 2P2 + Q2, then P2 = 2P3 + Q3, etc. finally expresses P as an integral linear combination of points Qi and of points whose height is bounded by a fixed constant chosen in advance: by the weak Mordell–Weil theorem and the second property of the height function P is thus expressed as an integral linear combination of a finite number of fixed points. The theorem however doesn't provide a method to determine any representatives of E(Q)/mE(Q). The <a href="/wiki/Rank_of_an_abelian_group" title="Rank of an abelian group">rank</a> of E(Q), that is the number of copies of Z in E(Q) or, equivalently, the number of independent points of infinite order, is called the rank of E. The <a href="/wiki/Birch_and_Swinnerton-Dyer_conjecture" title="Birch and Swinnerton-Dyer conjecture">Birch and Swinnerton-Dyer conjecture</a> is concerned with determining the rank. One conjectures that it can be arbitrarily large, even if only examples with relatively small rank are known. The elliptic curve with the currently largest exactly-known rank is <dl><dd>y2 + xy + y = x3 − x2 − 244537673336319601463803487168961769270757573821859853707x + 961710182053183034546222979258806817743270682028964434238957830989898438151121499931</dd></dl> It has rank 20, found by <a href="/wiki/Noam_Elkies" title="Noam Elkies">Noam Elkies</a> and Zev Klagsbrun in 2020. Curves of rank higher than 20 have been known since 1994, with lower bounds on their ranks ranging from 21 to 29, but their exact ranks are not known and in particular it is not proven which of them have higher rank than the others or which is the true "current champion".<a href="#cite_note-9">[9]</a> As for the groups constituting the <a href="/wiki/Torsion_subgroup" title="Torsion subgroup">torsion subgroup</a> of E(Q), the following is known:<a href="#cite_note-10">[10]</a> the torsion subgroup of E(Q) is one of the 15 following groups (<a href="/wiki/Mazur%27s_torsion_theorem" class="mw-redirect" title="Mazur's torsion theorem">a theorem</a> due to <a href="/wiki/Barry_Mazur" title="Barry Mazur">Barry Mazur</a>): Z/NZ for N = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or 12, or Z/2Z × Z/2NZ with N = 1, 2, 3, 4. Examples for every case are known. Moreover, elliptic curves whose Mordell–Weil groups over Q have the same torsion groups belong to a parametrized family.<a href="#cite_note-11">[11]</a> <div class="mw-heading mw-heading3"><h3 id="The_Birch_and_Swinnerton-Dyer_conjecture">The Birch and Swinnerton-Dyer conjecture</h3>[<a href="/w/index.php?title=Elliptic_curve&action=edit&section=8" title="Edit section: The Birch and Swinnerton-Dyer conjecture">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Birch_and_Swinnerton-Dyer_conjecture" title="Birch and Swinnerton-Dyer conjecture">Birch and Swinnerton-Dyer conjecture</a></div> The Birch and Swinnerton-Dyer conjecture (BSD) is one of the <a href="/wiki/Millennium_problem" class="mw-redirect" title="Millennium problem">Millennium problems</a> of the <a href="/wiki/Clay_Mathematics_Institute" title="Clay Mathematics Institute">Clay Mathematics Institute</a>. The conjecture relies on analytic and arithmetic objects defined by the elliptic curve in question. At the analytic side, an important ingredient is a function of a complex variable, L, the <a href="/wiki/Hasse%E2%80%93Weil_zeta_function" title="Hasse–Weil zeta function">Hasse–Weil zeta function</a> of E over Q. This function is a variant of the <a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a> and <a href="/wiki/Dirichlet_L-function" title="Dirichlet L-function">Dirichlet L-functions</a>. It is defined as an <a href="/wiki/Euler_product" title="Euler product">Euler product</a>, with one factor for every <a href="/wiki/Prime_number" title="Prime number">prime number</a> p. For a curve E over Q given by a minimal equation <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>x</mi> <mi>y</mi> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mi>y</mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4c91626d1e5a1e25d3e4e3be4eba70f8df713c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:40.674ex; height:3.009ex;" alt="{\displaystyle y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}}" /></dd></dl> with integral coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc77764b2e74e64a63341054fa90f3e07db275f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.029ex; height:2.009ex;" alt="{\displaystyle a_{i}}" />, reducing the coefficients <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modulo</a> p defines an elliptic curve over the <a href="/wiki/Finite_field" title="Finite field">finite field</a> Fp (except for a finite number of primes p, where the reduced curve has a <a href="/wiki/Mathematical_singularity" class="mw-redirect" title="Mathematical singularity">singularity</a> and thus fails to be elliptic, in which case E is said to be of <a href="/wiki/Bad_reduction" class="mw-redirect" title="Bad reduction">bad reduction</a> at p). The zeta function of an elliptic curve over a finite field Fp is, in some sense, a <a href="/wiki/Generating_function" title="Generating function">generating function</a> assembling the information of the number of points of E with values in the finite <a href="/wiki/Field_extension" title="Field extension">field extensions</a> Fpn of Fp. It is given by<a href="#cite_note-12">[12]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z(E(\mathbf {F} _{p}),T)=\exp \left(\sum _{n=1}^{\infty }\#\left[E({\mathbf {F} }_{p^{n}})\right]{\frac {T^{n}}{n}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi mathvariant="normal">#</mi> <mrow> <mo>[</mo> <mrow> <mi>E</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>n</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z(E(\mathbf {F} _{p}),T)=\exp \left(\sum _{n=1}^{\infty }\#\left[E({\mathbf {F} }_{p^{n}})\right]{\frac {T^{n}}{n}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5e3238b8c26886b9bb7ee84530036d9c50133ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:41.631ex; height:7.509ex;" alt="{\displaystyle Z(E(\mathbf {F} _{p}),T)=\exp \left(\sum _{n=1}^{\infty }\#\left[E({\mathbf {F} }_{p^{n}})\right]{\frac {T^{n}}{n}}\right)}" /></dd></dl> The interior sum of the exponential resembles the development of the <a href="/wiki/Logarithm" title="Logarithm">logarithm</a> and, in fact, the so-defined zeta function is a <a href="/wiki/Rational_function" title="Rational function">rational function</a> in T: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z(E(\mathbf {F} _{p}),T)={\frac {1-a_{p}T+pT^{2}}{(1-T)(1-pT)}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mi>T</mi> <mo>+</mo> <mi>p</mi> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z(E(\mathbf {F} _{p}),T)={\frac {1-a_{p}T+pT^{2}}{(1-T)(1-pT)}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e95b0cdb5a2f49c15d587b4f901d6e2bf534b9e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:33.134ex; height:6.676ex;" alt="{\displaystyle Z(E(\mathbf {F} _{p}),T)={\frac {1-a_{p}T+pT^{2}}{(1-T)(1-pT)}},}" /></dd></dl> where the 'trace of Frobenius' term<a href="#cite_note-13">[13]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3a5f11f304b2976ca9d33d1575c148e791728e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.289ex; height:2.343ex;" alt="{\displaystyle a_{p}}" /> is defined to be the difference between the 'expected' number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5885ec01d3b5670fd5f88847f32da2b3dd62f60c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.262ex; height:2.509ex;" alt="{\displaystyle p+1}" /> and the number of points on the elliptic curve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" /> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d35035371db7bee93733c68c1802114c17d8bb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.479ex; height:2.843ex;" alt="{\displaystyle \mathbb {F} _{p}}" />, viz. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{p}=p+1-\#E(\mathbb {F} _{p})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mi>p</mi> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mi mathvariant="normal">#</mi> <mi>E</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{p}=p+1-\#E(\mathbb {F} _{p})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c2c16dd51d81f442b929fcfac7f4349b1ca6829" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.4ex; height:3.009ex;" alt="{\displaystyle a_{p}=p+1-\#E(\mathbb {F} _{p})}" /></dd></dl> or equivalently, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \#E(\mathbb {F} _{p})=p+1-a_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">#</mi> <mi>E</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>p</mi> <mo>+</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \#E(\mathbb {F} _{p})=p+1-a_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2abd70f0cb4fdebafdddcded342075a3b077672b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.4ex; height:3.009ex;" alt="{\displaystyle \#E(\mathbb {F} _{p})=p+1-a_{p}}" />.</dd></dl> We may define the same quantities and functions over an arbitrary finite field of characteristic <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" />, with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=p^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q=p^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b95f6525025c9ae1432ea9bd50c3c1cab4b7c6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.556ex; height:2.676ex;" alt="{\displaystyle q=p^{n}}" /> replacing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" /> everywhere. The <a href="/wiki/Hasse%E2%80%93Weil_zeta_function#Example:_elliptic_curve_over_Q" title="Hasse–Weil zeta function">L-function</a> of E over Q is then defined by collecting this information together, for all primes p. It is defined by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(E(\mathbf {Q} ),s)=\prod _{p\not \mid N}\left(1-a_{p}p^{-s}+p^{1-2s}\right)^{-1}\cdot \prod _{p\mid N}\left(1-a_{p}p^{-s}\right)^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>∤</mo> <mi>N</mi> </mrow> </munder> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>s</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⋅</mo> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>∣</mo> <mi>N</mi> </mrow> </munder> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L(E(\mathbf {Q} ),s)=\prod _{p\not \mid N}\left(1-a_{p}p^{-s}+p^{1-2s}\right)^{-1}\cdot \prod _{p\mid N}\left(1-a_{p}p^{-s}\right)^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/649138012296e62b178c98b015d6df804a2bd59d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:58.947ex; height:6.343ex;" alt="{\displaystyle L(E(\mathbf {Q} ),s)=\prod _{p\not \mid N}\left(1-a_{p}p^{-s}+p^{1-2s}\right)^{-1}\cdot \prod _{p\mid N}\left(1-a_{p}p^{-s}\right)^{-1}}" /></dd></dl> where N is the <a href="/wiki/Conductor_of_an_elliptic_curve" title="Conductor of an elliptic curve">conductor</a> of E, i.e. the product of primes with bad reduction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Delta (E\mod p)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">(</mo> <mi>E</mi> <mspace width="1em"></mspace> <mi>mod</mi> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\Delta (E\mod p)=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9b0b4c43588eb6890f35c1aad8dc2ea5855c0da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.118ex; height:2.843ex;" alt="{\displaystyle (\Delta (E\mod p)=0}" />),<a href="#cite_note-14">[14]</a> in which case ap is defined differently from the method above: see Silverman (1986) below. For example <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E:y^{2}=x^{3}+14x+19}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>:</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>14</mn> <mi>x</mi> <mo>+</mo> <mn>19</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E:y^{2}=x^{3}+14x+19}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21efad3e344048541ec956c91c063b7cd886e862" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.07ex; height:3.009ex;" alt="{\displaystyle E:y^{2}=x^{3}+14x+19}" /> has bad reduction at 17, because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\mod 17:y^{2}=x^{3}-3x+2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mspace width="1em"></mspace> <mi>mod</mi> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mn>17</mn> <mo>:</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\mod 17:y^{2}=x^{3}-3x+2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19c1a1bad54a30bd2fc8bbc417c33305994613e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:31.332ex; height:3.009ex;" alt="{\displaystyle E\mod 17:y^{2}=x^{3}-3x+2}" /> has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta =0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf057da503668fa097746562ae91517330ce5b58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.197ex; height:2.176ex;" alt="{\displaystyle \Delta =0}" />. This product <a href="/wiki/Absolute_convergence" title="Absolute convergence">converges</a> for Re(s) > 3/2 only. Hasse's conjecture affirms that the L-function admits an <a href="/wiki/Analytic_continuation" title="Analytic continuation">analytic continuation</a> to the whole complex plane and satisfies a <a href="/wiki/Functional_equation" title="Functional equation">functional equation</a> relating, for any s, L(E, s) to L(E, 2 − s). In 1999 this was shown to be a consequence of the proof of the Shimura–Taniyama–Weil conjecture, which asserts that every elliptic curve over Q is a <a href="/wiki/Modular_curve" title="Modular curve">modular curve</a>, which implies that its L-function is the L-function of a <a href="/wiki/Modular_form" title="Modular form">modular form</a> whose analytic continuation is known. One can therefore speak about the values of L(E, s) at any complex number s. At s=1 (the conductor product can be discarded as it is finite), the L-function becomes <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(E(\mathbf {Q} ),1)=\prod _{p\not \mid N}\left(1-a_{p}p^{-1}+p^{-1}\right)^{-1}=\prod _{p\not \mid N}{\frac {p}{p-a_{p}+1}}=\prod _{p\not \mid N}{\frac {p}{\#E(\mathbb {F} _{p})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>∤</mo> <mi>N</mi> </mrow> </munder> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>∤</mo> <mi>N</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mrow> <mi>p</mi> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>∤</mo> <mi>N</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mrow> <mi mathvariant="normal">#</mi> <mi>E</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L(E(\mathbf {Q} ),1)=\prod _{p\not \mid N}\left(1-a_{p}p^{-1}+p^{-1}\right)^{-1}=\prod _{p\not \mid N}{\frac {p}{p-a_{p}+1}}=\prod _{p\not \mid N}{\frac {p}{\#E(\mathbb {F} _{p})}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95906e45865da7966cdd10b46fedf3712ea4c914" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:71.12ex; height:6.509ex;" alt="{\displaystyle L(E(\mathbf {Q} ),1)=\prod _{p\not \mid N}\left(1-a_{p}p^{-1}+p^{-1}\right)^{-1}=\prod _{p\not \mid N}{\frac {p}{p-a_{p}+1}}=\prod _{p\not \mid N}{\frac {p}{\#E(\mathbb {F} _{p})}}}" /></dd></dl> The Birch and Swinnerton-Dyer conjecture relates the arithmetic of the curve to the behaviour of this L-function at s = 1. It affirms that the vanishing order of the L-function at s = 1 equals the rank of E and predicts the leading term of the Laurent series of L(E, s) at that point in terms of several quantities attached to the elliptic curve. Much like the <a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a>, the truth of the BSD conjecture would have multiple consequences, including the following two: <ul><li>A <a href="/wiki/Congruent_number" title="Congruent number">congruent number</a> is defined as an odd <a href="/wiki/Square-free_integer" title="Square-free integer">square-free integer</a> n which is the area of a right triangle with rational side lengths. It is known that n is a congruent number if and only if the elliptic curve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y^{2}=x^{3}-n^{2}x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{2}=x^{3}-n^{2}x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6cb0d824151d70c0c4068b412047629e328eea9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.316ex; height:3.009ex;" alt="{\displaystyle y^{2}=x^{3}-n^{2}x}" /> has a rational point of infinite order; assuming BSD, this is equivalent to its L-function having a zero at s = 1. <a href="/wiki/Tunnell%27s_theorem" title="Tunnell's theorem">Tunnell</a> has shown a related result: assuming BSD, n is a congruent number if and only if the number of triplets of integers (x, y, z) satisfying <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2x^{2}+y^{2}+8z^{2}=n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>8</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2x^{2}+y^{2}+8z^{2}=n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54c8b35f5cdc3631daed2defab3c2e7cfcd9ddae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.242ex; height:3.009ex;" alt="{\displaystyle 2x^{2}+y^{2}+8z^{2}=n}" /> is twice the number of triples satisfying <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2x^{2}+y^{2}+32z^{2}=n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>32</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2x^{2}+y^{2}+32z^{2}=n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/279dcee89bedfb2849774016eb6242c3b624b41d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.405ex; height:3.009ex;" alt="{\displaystyle 2x^{2}+y^{2}+32z^{2}=n}" />. The interest in this statement is that the condition is easy to check.<a href="#cite_note-15">[15]</a></li> <li>In a different direction, certain analytic methods allow for an estimation of the order of zero in the center of the <a href="/wiki/Critical_strip" class="mw-redirect" title="Critical strip">critical strip</a> for certain L-functions. Admitting BSD, these estimations correspond to information about the rank of families of the corresponding elliptic curves. For example: assuming the <a href="/wiki/Generalized_Riemann_hypothesis" title="Generalized Riemann hypothesis">generalized Riemann hypothesis</a> and BSD, the average rank of curves given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y^{2}=x^{3}+ax+b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mi>a</mi> <mi>x</mi> <mo>+</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{2}=x^{3}+ax+b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dbe6cab1bc2c7f1c99757dc6e5d7a517cf9b4f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.935ex; height:3.009ex;" alt="{\displaystyle y^{2}=x^{3}+ax+b}" /> is smaller than 2.<a href="#cite_note-16">[16]</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Elliptic_curves_over_finite_fields">Elliptic curves over finite fields</h2>[<a href="/w/index.php?title=Elliptic_curve&action=edit&section=9" title="Edit section: Elliptic curves over finite fields">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Arithmetic_of_abelian_varieties" title="Arithmetic of abelian varieties">Arithmetic of abelian varieties</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Elliptic_curve_on_Z61.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Elliptic_curve_on_Z61.svg/260px-Elliptic_curve_on_Z61.svg.png" decoding="async" width="260" height="161" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Elliptic_curve_on_Z61.svg/390px-Elliptic_curve_on_Z61.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Elliptic_curve_on_Z61.svg/520px-Elliptic_curve_on_Z61.svg.png 2x" data-file-width="987" data-file-height="610" /></a><figcaption>Set of affine points of elliptic curve y2 = x3 − x over finite field F61.</figcaption></figure> Let K = Fq be the <a href="/wiki/Finite_field" title="Finite field">finite field</a> with q elements and E an elliptic curve defined over K. While the precise <a href="/wiki/Counting_points_on_elliptic_curves" title="Counting points on elliptic curves">number of rational points of an elliptic curve</a> E over K is in general difficult to compute, <a href="/wiki/Hasse%27s_theorem_on_elliptic_curves" title="Hasse's theorem on elliptic curves">Hasse's theorem on elliptic curves</a> gives the following inequality: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\#E(K)-(q+1)|\leq 2{\sqrt {q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">#</mi> <mi>E</mi> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>q</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\#E(K)-(q+1)|\leq 2{\sqrt {q}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/121d363add8fe7308e5fc031e62cfc79605f5dff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:25.869ex; height:3.176ex;" alt="{\displaystyle |\#E(K)-(q+1)|\leq 2{\sqrt {q}}}" /></dd></dl> In other words, the number of points on the curve grows proportionally to the number of elements in the field. This fact can be understood and proven with the help of some general theory; see <a href="/wiki/Local_zeta_function" title="Local zeta function">local zeta function</a> and <a href="/wiki/%C3%89tale_cohomology#An_application_to_curves" title="Étale cohomology">étale cohomology</a> for example. <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Elliptic_curve_on_Z89.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Elliptic_curve_on_Z89.svg/260px-Elliptic_curve_on_Z89.svg.png" decoding="async" width="260" height="161" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Elliptic_curve_on_Z89.svg/390px-Elliptic_curve_on_Z89.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Elliptic_curve_on_Z89.svg/520px-Elliptic_curve_on_Z89.svg.png 2x" data-file-width="987" data-file-height="610" /></a><figcaption>Set of affine points of elliptic curve y2 = x3 − x over finite field F89.</figcaption></figure> The set of points E(Fq) is a finite abelian group. It is always cyclic or the product of two cyclic groups. For example,<a href="#cite_note-17">[17]</a> the curve defined by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y^{2}=x^{3}-x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{2}=x^{3}-x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8f2e8f7d24923b97b74d1ce62bf524be635c514" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.867ex; height:3.009ex;" alt="{\displaystyle y^{2}=x^{3}-x}" /></dd></dl> over F71 has 72 points (71 <a href="/wiki/Affine_space#Affine_coordinates" title="Affine space">affine points</a> including (0,0) and one <a href="/wiki/Point_at_infinity" title="Point at infinity">point at infinity</a>) over this field, whose group structure is given by Z/2Z × Z/36Z. The number of points on a specific curve can be computed with <a href="/wiki/Schoof%27s_algorithm" title="Schoof's algorithm">Schoof's algorithm</a>. <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Elliptic_curve_on_Z71.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/Elliptic_curve_on_Z71.svg/330px-Elliptic_curve_on_Z71.svg.png" decoding="async" width="330" height="204" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/Elliptic_curve_on_Z71.svg/495px-Elliptic_curve_on_Z71.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/49/Elliptic_curve_on_Z71.svg/660px-Elliptic_curve_on_Z71.svg.png 2x" data-file-width="987" data-file-height="610" /></a><figcaption>Set of affine points of elliptic curve y2 = x3 − x over finite field F71.</figcaption></figure> Studying the curve over the <a href="/wiki/Field_extension" title="Field extension">field extensions</a> of Fq is facilitated by the introduction of the local zeta function of E over Fq, defined by a generating series (also see above) <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z(E(K),T)=\exp \left(\sum _{n=1}^{\infty }\#\left[E(K_{n})\right]{T^{n} \over n}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi mathvariant="normal">#</mi> <mrow> <mo>[</mo> <mrow> <mi>E</mi> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>n</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z(E(K),T)=\exp \left(\sum _{n=1}^{\infty }\#\left[E(K_{n})\right]{T^{n} \over n}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/608830007ce25cbc3a81ef246adff37a4bf9572e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:40.44ex; height:7.509ex;" alt="{\displaystyle Z(E(K),T)=\exp \left(\sum _{n=1}^{\infty }\#\left[E(K_{n})\right]{T^{n} \over n}\right)}" /></dd></dl> where the field Kn is the (unique up to isomorphism) extension of K = Fq of degree n (that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{n}=F_{q^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{n}=F_{q^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a78cad194284cae4362d87ebdd14c4590e62435" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.745ex; height:2.843ex;" alt="{\displaystyle K_{n}=F_{q^{n}}}" />). The zeta function is a rational function in T. To see this, consider the integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" /> such that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \#E(K)=1-a+q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">#</mi> <mi>E</mi> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>a</mi> <mo>+</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \#E(K)=1-a+q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b753617be8b09381ef85bdbed31d80f805924b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.828ex; height:2.843ex;" alt="{\displaystyle \#E(K)=1-a+q}" /></dd></dl> There is a complex number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /> such that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-a+q=(1-\alpha )(1-{\bar {\alpha }})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mi>a</mi> <mo>+</mo> <mi>q</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-a+q=(1-\alpha )(1-{\bar {\alpha }})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56df9240e3d7d016f10046b16a1a69b17003d5ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.841ex; height:2.843ex;" alt="{\displaystyle 1-a+q=(1-\alpha )(1-{\bar {\alpha }})}" /></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\alpha }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {\alpha }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef29452eb836fce0a8544d63f3a8f76fccc74ece" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:2.009ex;" alt="{\displaystyle {\bar {\alpha }}}" /> is the <a href="/wiki/Complex_conjugate" title="Complex conjugate">complex conjugate</a>, and so we have <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha +{\bar {\alpha }}=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha +{\bar {\alpha }}=a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bff5b0df620fafbe87e998291c82f00e2a57374" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.144ex; height:2.176ex;" alt="{\displaystyle \alpha +{\bar {\alpha }}=a}" /></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha {\bar {\alpha }}=q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha {\bar {\alpha }}=q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ec11fbd3f107670f8e378f11fbd977ea6a42834" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.143ex; height:2.343ex;" alt="{\displaystyle \alpha {\bar {\alpha }}=q}" /></dd></dl> We choose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /> so that its <a href="/wiki/Absolute_value" title="Absolute value">absolute value</a> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>q</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {q}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fec68a26ee361e6e7450c74fadea8334650ec369" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.005ex; height:3.009ex;" alt="{\displaystyle {\sqrt {q}}}" />, that is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =q^{\frac {1}{2}}e^{i\theta },{\bar {\alpha }}=q^{\frac {1}{2}}e^{-i\theta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>=</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>θ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =q^{\frac {1}{2}}e^{i\theta },{\bar {\alpha }}=q^{\frac {1}{2}}e^{-i\theta }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/376e61bb90ab203bac298b3305bc09781957a565" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.423ex; height:3.843ex;" alt="{\displaystyle \alpha =q^{\frac {1}{2}}e^{i\theta },{\bar {\alpha }}=q^{\frac {1}{2}}e^{-i\theta }}" />, and that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \theta ={\frac {a}{2{\sqrt {q}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>q</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \theta ={\frac {a}{2{\sqrt {q}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/945950cd90a49e4aa4994fbbc540af09f50951c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:12.691ex; height:5.676ex;" alt="{\displaystyle \cos \theta ={\frac {a}{2{\sqrt {q}}}}}" />. Note that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |a|\leq 2{\sqrt {q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>q</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |a|\leq 2{\sqrt {q}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8337e2807883f825c162e5a7c0276e7b32c1bfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:9.79ex; height:3.176ex;" alt="{\displaystyle |a|\leq 2{\sqrt {q}}}" />. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /> can then be used in the local zeta function as its values when raised to the various powers of n can be said to reasonably approximate the behaviour of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/790f9209748c2dca7ed7b81932c37c02af1dbc31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}" />, in that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \#E(K_{n})=1-a_{n}+q^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">#</mi> <mi>E</mi> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \#E(K_{n})=1-a_{n}+q^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3355d0d37947bd13ff0b59d90a74fbfb045f7057" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.4ex; height:2.843ex;" alt="{\displaystyle \#E(K_{n})=1-a_{n}+q^{n}}" /></dd></dl> Using the <a href="/wiki/List_of_logarithmic_identities#Series_representation" title="List of logarithmic identities">Taylor series for the natural logarithm</a>, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{alignedat}{2}Z(E(K),T)&=\exp \left(\sum _{n=1}^{\infty }\left(1-\alpha ^{n}-{\bar {\alpha }}^{n}+q^{n}\right){T^{n} \over n}\right)\\&=\exp \left(\sum _{n=1}^{\infty }{T^{n} \over n}-\sum _{n=1}^{\infty }\alpha ^{n}{T^{n} \over n}-\sum _{n=1}^{\infty }{\bar {\alpha }}^{n}{T^{n} \over n}+\sum _{n=1}^{\infty }q^{n}{T^{n} \over n}\right)\\&=\exp \left(-\ln(1-T)+\ln(1-\alpha T)+\ln(1-{\bar {\alpha }}T)-\ln(1-qT)\right)\\&=\exp \left(\ln {\frac {(1-\alpha T)(1-{\bar {\alpha }}T)}{(1-T)(1-qT)}}\right)\\&={\frac {(1-\alpha T)(1-{\bar {\alpha }}T)}{(1-T)(1-qT)}}\\\end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <mi>Z</mi> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>n</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>n</mi> </mfrac> </mrow> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>n</mi> </mfrac> </mrow> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>n</mi> </mfrac> </mrow> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>n</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mi>T</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mi>T</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>q</mi> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mi>T</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>q</mi> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mi>T</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>q</mi> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{2}Z(E(K),T)&=\exp \left(\sum _{n=1}^{\infty }\left(1-\alpha ^{n}-{\bar {\alpha }}^{n}+q^{n}\right){T^{n} \over n}\right)\\&=\exp \left(\sum _{n=1}^{\infty }{T^{n} \over n}-\sum _{n=1}^{\infty }\alpha ^{n}{T^{n} \over n}-\sum _{n=1}^{\infty }{\bar {\alpha }}^{n}{T^{n} \over n}+\sum _{n=1}^{\infty }q^{n}{T^{n} \over n}\right)\\&=\exp \left(-\ln(1-T)+\ln(1-\alpha T)+\ln(1-{\bar {\alpha }}T)-\ln(1-qT)\right)\\&=\exp \left(\ln {\frac {(1-\alpha T)(1-{\bar {\alpha }}T)}{(1-T)(1-qT)}}\right)\\&={\frac {(1-\alpha T)(1-{\bar {\alpha }}T)}{(1-T)(1-qT)}}\\\end{alignedat}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f517e30c4b44bdce312bfb018faf17d16c8b8a1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -14.912ex; margin-bottom: -0.26ex; width:73.335ex; height:31.509ex;" alt="{\displaystyle {\begin{alignedat}{2}Z(E(K),T)&=\exp \left(\sum _{n=1}^{\infty }\left(1-\alpha ^{n}-{\bar {\alpha }}^{n}+q^{n}\right){T^{n} \over n}\right)\\&=\exp \left(\sum _{n=1}^{\infty }{T^{n} \over n}-\sum _{n=1}^{\infty }\alpha ^{n}{T^{n} \over n}-\sum _{n=1}^{\infty }{\bar {\alpha }}^{n}{T^{n} \over n}+\sum _{n=1}^{\infty }q^{n}{T^{n} \over n}\right)\\&=\exp \left(-\ln(1-T)+\ln(1-\alpha T)+\ln(1-{\bar {\alpha }}T)-\ln(1-qT)\right)\\&=\exp \left(\ln {\frac {(1-\alpha T)(1-{\bar {\alpha }}T)}{(1-T)(1-qT)}}\right)\\&={\frac {(1-\alpha T)(1-{\bar {\alpha }}T)}{(1-T)(1-qT)}}\\\end{alignedat}}}" /></dd></dl> Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1-\alpha T)(1-{\bar {\alpha }}T)=1-aT+qT^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mi>T</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>a</mi> <mi>T</mi> <mo>+</mo> <mi>q</mi> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1-\alpha T)(1-{\bar {\alpha }}T)=1-aT+qT^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6a775e16d044f0de9ee374efce42f756def5a12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.524ex; height:3.176ex;" alt="{\displaystyle (1-\alpha T)(1-{\bar {\alpha }}T)=1-aT+qT^{2}}" />, so finally <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z(E(K),T)={\frac {1-aT+qT^{2}}{(1-qT)(1-T)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>a</mi> <mi>T</mi> <mo>+</mo> <mi>q</mi> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>q</mi> <mi>T</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z(E(K),T)={\frac {1-aT+qT^{2}}{(1-qT)(1-T)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5589ace1585c89c93a95506e44d382296594cc87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:31.712ex; height:6.509ex;" alt="{\displaystyle Z(E(K),T)={\frac {1-aT+qT^{2}}{(1-qT)(1-T)}}}" /></dd></dl> For example,<a href="#cite_note-18">[18]</a> the zeta function of E : y2 + y = x3 over the field F2 is given by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1+2T^{2}}{(1-T)(1-2T)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mn>2</mn> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1+2T^{2}}{(1-T)(1-2T)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1acb11b31480f8423e0200145ae90e1ac3aacc6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:16.895ex; height:6.509ex;" alt="{\displaystyle {\frac {1+2T^{2}}{(1-T)(1-2T)}}}" /></dd></dl> which follows from: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|E(\mathbf {F} _{2^{r}})\right|={\begin{cases}2^{r}+1&r{\text{ odd}}\\2^{r}+1-2(-2)^{\frac {r}{2}}&r{\text{ even}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <mi>E</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mtd> <mtd> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> odd</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>r</mi> <mn>2</mn> </mfrac> </mrow> </msup> </mtd> <mtd> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> even</mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|E(\mathbf {F} _{2^{r}})\right|={\begin{cases}2^{r}+1&r{\text{ odd}}\\2^{r}+1-2(-2)^{\frac {r}{2}}&r{\text{ even}}\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22a6ad0890f134ba5d6abb79f2e9440ff953f7e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:39.169ex; height:6.509ex;" alt="{\displaystyle \left|E(\mathbf {F} _{2^{r}})\right|={\begin{cases}2^{r}+1&r{\text{ odd}}\\2^{r}+1-2(-2)^{\frac {r}{2}}&r{\text{ even}}\end{cases}}}" /></dd></dl> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q=2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26622af6012fb982cab4e9584f57dd4f364233b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.33ex; height:2.509ex;" alt="{\displaystyle q=2}" />, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |E|=2^{1}+1=3=1-a+2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>3</mn> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>a</mi> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |E|=2^{1}+1=3=1-a+2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbf034cc5c633a6cdc8f18b0b6934f4a6fb620b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.982ex; height:3.176ex;" alt="{\displaystyle |E|=2^{1}+1=3=1-a+2}" />, so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90d476e5e765a5d77bbcff32e4584579207ec7d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a=0}" />. The <a href="/wiki/Functional_equation" title="Functional equation">functional equation</a> is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z\left(E(K),{\frac {1}{qT}}\right)={\frac {1-a{\frac {1}{qT}}+q\left({\frac {1}{qT}}\right)^{2}}{(1-q{\frac {1}{qT}})(1-{\frac {1}{qT}})}}={\frac {q^{2}T^{2}-aqT+q}{(qT-q)(qT-1)}}=Z(E(K),T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mrow> <mo>(</mo> <mrow> <mi>E</mi> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>q</mi> <mi>T</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>q</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>q</mi> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>q</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>q</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>q</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>a</mi> <mi>q</mi> <mi>T</mi> <mo>+</mo> <mi>q</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mi>T</mi> <mo>−</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mi>T</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>Z</mi> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z\left(E(K),{\frac {1}{qT}}\right)={\frac {1-a{\frac {1}{qT}}+q\left({\frac {1}{qT}}\right)^{2}}{(1-q{\frac {1}{qT}})(1-{\frac {1}{qT}})}}={\frac {q^{2}T^{2}-aqT+q}{(qT-q)(qT-1)}}=Z(E(K),T)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81aac88dd625b7908168591bf5a8c2aa549497bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:73.908ex; height:9.843ex;" alt="{\displaystyle Z\left(E(K),{\frac {1}{qT}}\right)={\frac {1-a{\frac {1}{qT}}+q\left({\frac {1}{qT}}\right)^{2}}{(1-q{\frac {1}{qT}})(1-{\frac {1}{qT}})}}={\frac {q^{2}T^{2}-aqT+q}{(qT-q)(qT-1)}}=Z(E(K),T)}" /></dd></dl> As we are only interested in the behaviour of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/790f9209748c2dca7ed7b81932c37c02af1dbc31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}" />, we can use a reduced zeta function <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z(a,T)=\exp \left(\sum _{n=1}^{\infty }-a_{n}{T^{n} \over n}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>n</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z(a,T)=\exp \left(\sum _{n=1}^{\infty }-a_{n}{T^{n} \over n}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74e1bbdbc7c5d4c9abac120cbda9c03ce632ef52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:29.495ex; height:7.509ex;" alt="{\displaystyle Z(a,T)=\exp \left(\sum _{n=1}^{\infty }-a_{n}{T^{n} \over n}\right)}" /></dd></dl> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z(a,T)=\exp \left(\sum _{n=1}^{\infty }-\alpha ^{n}{T^{n} \over n}-{\bar {\alpha }}^{n}{T^{n} \over n}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mo>−</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>n</mi> </mfrac> </mrow> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>n</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z(a,T)=\exp \left(\sum _{n=1}^{\infty }-\alpha ^{n}{T^{n} \over n}-{\bar {\alpha }}^{n}{T^{n} \over n}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf85f7170bfc3eaf8e78abd9c6d583c27f78ec4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:39.074ex; height:7.509ex;" alt="{\displaystyle Z(a,T)=\exp \left(\sum _{n=1}^{\infty }-\alpha ^{n}{T^{n} \over n}-{\bar {\alpha }}^{n}{T^{n} \over n}\right)}" /></dd></dl> and so <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z(a,T)=\exp \left(\ln(1-\alpha T)+\ln(1-{\bar {\alpha }}T)\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mi>T</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z(a,T)=\exp \left(\ln(1-\alpha T)+\ln(1-{\bar {\alpha }}T)\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41b8c3669ec002a9c71ef64fe8f3f5135f242a3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.441ex; height:2.843ex;" alt="{\displaystyle Z(a,T)=\exp \left(\ln(1-\alpha T)+\ln(1-{\bar {\alpha }}T)\right)}" /></dd></dl> which leads directly to the local L-functions <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(E(K),T)=1-aT+qT^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>a</mi> <mi>T</mi> <mo>+</mo> <mi>q</mi> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L(E(K),T)=1-aT+qT^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20ed3f2a2da6a668680b56682e7b4473f4ee928d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.365ex; height:3.176ex;" alt="{\displaystyle L(E(K),T)=1-aT+qT^{2}}" /></dd></dl> The <a href="/wiki/Sato%E2%80%93Tate_conjecture" title="Sato–Tate conjecture">Sato–Tate conjecture</a> is a statement about how the error term <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2{\sqrt {q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>q</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2{\sqrt {q}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56d326d348e308c3ad44dc7f67c9220d25f44579" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.168ex; height:3.009ex;" alt="{\displaystyle 2{\sqrt {q}}}" /> in Hasse's theorem varies with the different primes q, if an elliptic curve E over Q is reduced modulo q. It was proven (for almost all such curves) in 2006 due to the results of Taylor, Harris and Shepherd-Barron,<a href="#cite_note-19">[19]</a> and says that the error terms are equidistributed. Elliptic curves over finite fields are notably applied in <a href="/wiki/Cryptography" title="Cryptography">cryptography</a> and for the <a href="/wiki/Factorization" title="Factorization">factorization</a> of large integers. These algorithms often make use of the group structure on the points of E. Algorithms that are applicable to general groups, for example the group of invertible elements in finite fields, F*q, can thus be applied to the group of points on an elliptic curve. For example, the <a href="/wiki/Discrete_logarithm" title="Discrete logarithm">discrete logarithm</a> is such an algorithm. The interest in this is that choosing an elliptic curve allows for more flexibility than choosing q (and thus the group of units in Fq). Also, the group structure of elliptic curves is generally more complicated. <div class="mw-heading mw-heading2"><h2 id="Elliptic_curves_over_a_general_field">Elliptic curves over a general field</h2>[<a href="/w/index.php?title=Elliptic_curve&action=edit&section=10" title="Edit section: Elliptic curves over a general field">edit</a>]</div> Elliptic curves can be defined over any <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> K; the formal definition of an elliptic curve is a non-singular projective algebraic curve over K with <a href="/wiki/Genus_(mathematics)" title="Genus (mathematics)">genus</a> 1 and endowed with a distinguished point defined over K. If the <a href="/wiki/Characteristic_(algebra)" title="Characteristic (algebra)">characteristic</a> of K is neither 2 nor 3, then every elliptic curve over K can be written in the form <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y^{2}=x^{3}-px-q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mi>p</mi> <mi>x</mi> <mo>−</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{2}=x^{3}-px-q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbf9d4b53128d0976ecf7852b328f032a2b04123" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.947ex; height:3.009ex;" alt="{\displaystyle y^{2}=x^{3}-px-q}" /></dd></dl> after a linear change of variables. Here p and q are elements of K such that the right hand side polynomial x3 − px − q does not have any double roots. If the characteristic is 2 or 3, then more terms need to be kept: in characteristic 3, the most general equation is of the form <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y^{2}=4x^{3}+b_{2}x^{2}+2b_{4}x+b_{6}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>4</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{2}=4x^{3}+b_{2}x^{2}+2b_{4}x+b_{6}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b464c38ebedff3bdd5fbfdf33c2f6d79163ced9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.412ex; height:3.009ex;" alt="{\displaystyle y^{2}=4x^{3}+b_{2}x^{2}+2b_{4}x+b_{6}}" /></dd></dl> for arbitrary constants b2, b4, b6 such that the polynomial on the right-hand side has distinct roots (the notation is chosen for historical reasons). In characteristic 2, even this much is not possible, and the most general equation is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>x</mi> <mi>y</mi> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mi>y</mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4c91626d1e5a1e25d3e4e3be4eba70f8df713c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:40.674ex; height:3.009ex;" alt="{\displaystyle y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}}" /></dd></dl> provided that the variety it defines is non-singular. If characteristic were not an obstruction, each equation would reduce to the previous ones by a suitable linear change of variables. One typically takes the curve to be the set of all points (x,y) which satisfy the above equation and such that both x and y are elements of the <a href="/wiki/Algebraic_closure" title="Algebraic closure">algebraic closure</a> of K. Points of the curve whose coordinates both belong to K are called K-rational points. Many of the preceding results remain valid when the field of definition of E is a <a href="/wiki/Number_field" class="mw-redirect" title="Number field">number field</a> K, that is to say, a finite <a href="/wiki/Field_extension" title="Field extension">field extension</a> of Q. In particular, the group E(K) of K-rational points of an elliptic curve E defined over K is finitely generated, which generalizes the Mordell–Weil theorem above. A theorem due to <a href="/wiki/Lo%C3%AFc_Merel" title="Loïc Merel">Loïc Merel</a> shows that for a given integer d, there are (<a href="/wiki/Up_to" title="Up to">up to</a> isomorphism) only finitely many groups that can occur as the torsion groups of E(K) for an elliptic curve defined over a number field K of <a href="/wiki/Degree_of_a_field_extension" title="Degree of a field extension">degree</a> d. More precisely,<a href="#cite_note-20">[20]</a> there is a number B(d) such that for any elliptic curve E defined over a number field K of degree d, any torsion point of E(K) is of <a href="/wiki/Order_(group_theory)" title="Order (group theory)">order</a> less than B(d). The theorem is effective: for d > 1, if a torsion point is of order p, with p prime, then <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p<d^{3d^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo><</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p<d^{3d^{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81c6452e6068111976c7d67925f0124f348aca6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:8.323ex; height:3.343ex;" alt="{\displaystyle p<d^{3d^{2}}}" /></dd></dl> As for the integral points, Siegel's theorem generalizes to the following: Let E be an elliptic curve defined over a number field K, x and y the Weierstrass coordinates. Then there are only finitely many points of E(K) whose x-coordinate is in the <a href="/wiki/Ring_of_integers" title="Ring of integers">ring of integers</a> OK. The properties of the Hasse–Weil zeta function and the Birch and Swinnerton-Dyer conjecture can also be extended to this more general situation. <div class="mw-heading mw-heading2"><h2 id="Elliptic_curves_over_the_complex_numbers">Elliptic curves over the complex numbers</h2>[<a href="/w/index.php?title=Elliptic_curve&action=edit&section=11" title="Edit section: Elliptic curves over the complex numbers">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Complex_multiplication" title="Complex multiplication">Complex multiplication</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Lattice_torsion_points.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d5/Lattice_torsion_points.svg/260px-Lattice_torsion_points.svg.png" decoding="async" width="260" height="325" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d5/Lattice_torsion_points.svg/390px-Lattice_torsion_points.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d5/Lattice_torsion_points.svg/520px-Lattice_torsion_points.svg.png 2x" data-file-width="800" data-file-height="1000" /></a><figcaption>An elliptic curve over the complex numbers is obtained as a quotient of the complex plane by a lattice Λ, here spanned by two fundamental periods ω1 and ω2. The four-torsion is also shown, corresponding to the lattice <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" />⁠1/4⁠Λ containing Λ.</figcaption></figure> The formulation of elliptic curves as the embedding of a <a href="/wiki/Torus" title="Torus">torus</a> in the <a href="/wiki/Complex_projective_plane" title="Complex projective plane">complex projective plane</a> follows naturally from a curious property of <a href="/wiki/Weierstrass%27s_elliptic_functions" class="mw-redirect" title="Weierstrass's elliptic functions">Weierstrass's elliptic functions</a>. These functions and their first derivative are related by the formula <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wp '(z)^{2}=4\wp (z)^{3}-g_{2}\wp (z)-g_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">℘</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>4</mn> <mi mathvariant="normal">℘</mi> <mo stretchy="false">(</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi mathvariant="normal">℘</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wp '(z)^{2}=4\wp (z)^{3}-g_{2}\wp (z)-g_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/591fdb589a978562eabce740e3baa8c2e9437c7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.189ex; height:3.176ex;" alt="{\displaystyle \wp '(z)^{2}=4\wp (z)^{3}-g_{2}\wp (z)-g_{3}}" /></dd></dl> Here, g2 and g3 are constants; ℘(z) is the <a href="/wiki/Weierstrass_elliptic_function" title="Weierstrass elliptic function">Weierstrass elliptic function</a> and ℘′(z) its derivative. It should be clear that this relation is in the form of an elliptic curve (over the <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>). The Weierstrass functions are doubly periodic; that is, they are <a href="/wiki/Fundamental_pair_of_periods" title="Fundamental pair of periods">periodic</a> with respect to a <a href="/wiki/Lattice_(group)" title="Lattice (group)">lattice</a> Λ; in essence, the Weierstrass functions are naturally defined on a torus T = C/Λ. This torus may be embedded in the complex projective plane by means of the map <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\mapsto \left[1:\wp (z):{\tfrac {1}{2}}\wp '(z)\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo stretchy="false">↦</mo> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mo>:</mo> <mi mathvariant="normal">℘</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mi mathvariant="normal">℘</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\mapsto \left[1:\wp (z):{\tfrac {1}{2}}\wp '(z)\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c351bf351c830acec700d97d49f2eeff421f8644" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:22.773ex; height:3.509ex;" alt="{\displaystyle z\mapsto \left[1:\wp (z):{\tfrac {1}{2}}\wp '(z)\right]}" /></dd></dl> This map is a <a href="/wiki/Group_isomorphism" title="Group isomorphism">group isomorphism</a> of the torus (considered with its natural group structure) with the chord-and-tangent group law on the cubic curve which is the image of this map. It is also an isomorphism of <a href="/wiki/Riemann_surface" title="Riemann surface">Riemann surfaces</a> from the torus to the cubic curve, so topologically, an elliptic curve is a torus. If the lattice Λ is related by multiplication by a non-zero complex number c to a lattice cΛ, then the corresponding curves are isomorphic. Isomorphism classes of elliptic curves are specified by the <a href="/wiki/J-invariant" title="J-invariant">j-invariant</a>. The isomorphism classes can be understood in a simpler way as well. The constants g2 and g3, called the <a href="/wiki/J-invariant" title="J-invariant">modular invariants</a>, are uniquely determined by the lattice, that is, by the structure of the torus. However, all real polynomials factorize completely into linear factors over the complex numbers, since the field of complex numbers is the <a href="/wiki/Algebraic_closure" title="Algebraic closure">algebraic closure</a> of the reals. So, the elliptic curve may be written as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y^{2}=x(x-1)(x-\lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{2}=x(x-1)(x-\lambda )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6beb85949e2737f6f14a2450459ce2512bf8a2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.119ex; height:3.176ex;" alt="{\displaystyle y^{2}=x(x-1)(x-\lambda )}" /></dd></dl> One finds that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}g_{2}'&={\frac {\sqrt[{3}]{4}}{3}}\left(\lambda ^{2}-\lambda +1\right)\\[4pt]g_{3}'&={\frac {1}{27}}(\lambda +1)\left(2\lambda ^{2}-5\lambda +2\right)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.7em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msubsup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mo>′</mo> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mroot> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> <mn>3</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>λ</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mo>′</mo> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>27</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>5</mn> <mi>λ</mi> <mo>+</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}g_{2}'&={\frac {\sqrt[{3}]{4}}{3}}\left(\lambda ^{2}-\lambda +1\right)\\[4pt]g_{3}'&={\frac {1}{27}}(\lambda +1)\left(2\lambda ^{2}-5\lambda +2\right)\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bf4038a9f1589968852c135e3541865cfb05136" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.319ex; margin-bottom: -0.186ex; width:31.791ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}g_{2}'&={\frac {\sqrt[{3}]{4}}{3}}\left(\lambda ^{2}-\lambda +1\right)\\[4pt]g_{3}'&={\frac {1}{27}}(\lambda +1)\left(2\lambda ^{2}-5\lambda +2\right)\end{aligned}}}" /></dd></dl> and <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j(\tau )=1728{\frac {{g_{2}'}^{3}}{{g_{2}'}^{3}-27{g_{3}'}^{2}}}=256{\frac {\left(\lambda ^{2}-\lambda +1\right)^{3}}{\lambda ^{2}\left(\lambda -1\right)^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1728</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mo>′</mo> </msubsup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mo>′</mo> </msubsup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>27</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mo>′</mo> </msubsup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>256</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>λ</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow> <mi>λ</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j(\tau )=1728{\frac {{g_{2}'}^{3}}{{g_{2}'}^{3}-27{g_{3}'}^{2}}}=256{\frac {\left(\lambda ^{2}-\lambda +1\right)^{3}}{\lambda ^{2}\left(\lambda -1\right)^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/531c9344a21dc65246f27a52084012f5f3e7a89d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; margin-left: -0.027ex; width:45.395ex; height:7.843ex;" alt="{\displaystyle j(\tau )=1728{\frac {{g_{2}'}^{3}}{{g_{2}'}^{3}-27{g_{3}'}^{2}}}=256{\frac {\left(\lambda ^{2}-\lambda +1\right)^{3}}{\lambda ^{2}\left(\lambda -1\right)^{2}}}}" /></dd></dl> with <a href="/wiki/J-invariant" title="J-invariant">j-invariant</a> j(τ) and λ(τ) is sometimes called the <a href="/wiki/Modular_lambda_function" title="Modular lambda function">modular lambda function</a>. For example, let τ = 2i, then λ(2i) = (−1 + √2)4 which implies g′2, g′3, and therefore g′23 − 27g′32 of the formula above are all <a href="/wiki/Algebraic_numbers" class="mw-redirect" title="Algebraic numbers">algebraic numbers</a> if τ involves an <a href="/wiki/Imaginary_quadratic_field" class="mw-redirect" title="Imaginary quadratic field">imaginary quadratic field</a>. In fact, it yields the integer j(2i) = 663 = 287496. In contrast, the <a href="/wiki/Modular_discriminant" class="mw-redirect" title="Modular discriminant">modular discriminant</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta (\tau )=g_{2}(\tau )^{3}-27g_{3}(\tau )^{2}=(2\pi )^{12}\,\eta ^{24}(\tau )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>τ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>27</mn> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>τ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msup> <mspace width="thinmathspace"></mspace> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>24</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta (\tau )=g_{2}(\tau )^{3}-27g_{3}(\tau )^{2}=(2\pi )^{12}\,\eta ^{24}(\tau )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/175a7909bb1bf1a87e1a78b1adbb4a90cbd399e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.395ex; height:3.176ex;" alt="{\displaystyle \Delta (\tau )=g_{2}(\tau )^{3}-27g_{3}(\tau )^{2}=(2\pi )^{12}\,\eta ^{24}(\tau )}" /></dd></dl> is generally a <a href="/wiki/Transcendental_number" title="Transcendental number">transcendental number</a>. In particular, the value of the <a href="/wiki/Dedekind_eta_function#Special_values" title="Dedekind eta function">Dedekind eta function</a> η(2i) is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta (2i)={\frac {\Gamma \left({\frac {1}{4}}\right)}{2^{\frac {11}{8}}\pi ^{\frac {3}{4}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>i</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Γ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>11</mn> <mn>8</mn> </mfrac> </mrow> </msup> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta (2i)={\frac {\Gamma \left({\frac {1}{4}}\right)}{2^{\frac {11}{8}}\pi ^{\frac {3}{4}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dee379130c0d5d896dd42240f595fc2e575f67b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:15.514ex; height:9.009ex;" alt="{\displaystyle \eta (2i)={\frac {\Gamma \left({\frac {1}{4}}\right)}{2^{\frac {11}{8}}\pi ^{\frac {3}{4}}}}}" /></dd></dl> Note that the <a href="/wiki/Uniformization_theorem" title="Uniformization theorem">uniformization theorem</a> implies that every <a href="/wiki/Compact_space" title="Compact space">compact</a> Riemann surface of genus one can be represented as a torus. This also allows an easy understanding of the <a href="/wiki/Torsion_subgroup" title="Torsion subgroup">torsion points</a> on an elliptic curve: if the lattice Λ is spanned by the fundamental periods ω1 and ω2, then the n-torsion points are the (equivalence classes of) points of the form <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {a}{n}}\omega _{1}+{\frac {b}{n}}\omega _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>n</mi> </mfrac> </mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>n</mi> </mfrac> </mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a}{n}}\omega _{1}+{\frac {b}{n}}\omega _{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2abe8a80b1940babf08a836cd45278de7aaf6ba6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.302ex; height:5.343ex;" alt="{\displaystyle {\frac {a}{n}}\omega _{1}+{\frac {b}{n}}\omega _{2}}" /></dd></dl> for integers a and b in the range 0 ≤ (a, b) < n. If <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E:y^{2}=4(x-e_{1})(x-e_{2})(x-e_{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>:</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>4</mn> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E:y^{2}=4(x-e_{1})(x-e_{2})(x-e_{3})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d8f75e9c68cfdc45bd9a484b358f7e3361469f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.54ex; height:3.176ex;" alt="{\displaystyle E:y^{2}=4(x-e_{1})(x-e_{2})(x-e_{3})}" /></dd></dl> is an elliptic curve over the complex numbers and <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}={\sqrt {e_{1}-e_{3}}},\qquad b_{0}={\sqrt {e_{1}-e_{2}}},\qquad c_{0}={\sqrt {e_{2}-e_{3}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </msqrt> </mrow> <mo>,</mo> <mspace width="2em"></mspace> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </msqrt> </mrow> <mo>,</mo> <mspace width="2em"></mspace> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}={\sqrt {e_{1}-e_{3}}},\qquad b_{0}={\sqrt {e_{1}-e_{2}}},\qquad c_{0}={\sqrt {e_{2}-e_{3}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb9a1b615f4904dc4194e08b43858555f73d3135" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:54.853ex; height:3.009ex;" alt="{\displaystyle a_{0}={\sqrt {e_{1}-e_{3}}},\qquad b_{0}={\sqrt {e_{1}-e_{2}}},\qquad c_{0}={\sqrt {e_{2}-e_{3}}},}" /></dd></dl> then a pair of fundamental periods of E can be calculated very rapidly by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{1}={\frac {\pi }{\operatorname {M} (a_{0},b_{0})}},\qquad \omega _{2}={\frac {\pi }{\operatorname {M} (c_{0},ib_{0})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mrow> <mi mathvariant="normal">M</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="2em"></mspace> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mrow> <mi mathvariant="normal">M</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mi>i</mi> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{1}={\frac {\pi }{\operatorname {M} (a_{0},b_{0})}},\qquad \omega _{2}={\frac {\pi }{\operatorname {M} (c_{0},ib_{0})}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/693276e9c1fa5ceac4cdbc249bed5c6d88673fc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:37.748ex; height:5.509ex;" alt="{\displaystyle \omega _{1}={\frac {\pi }{\operatorname {M} (a_{0},b_{0})}},\qquad \omega _{2}={\frac {\pi }{\operatorname {M} (c_{0},ib_{0})}}}" /></dd></dl> M(w, z) is the <a href="/wiki/Arithmetic%E2%80%93geometric_mean" title="Arithmetic–geometric mean">arithmetic–geometric mean</a> of w and z. At each step of the arithmetic–geometric mean iteration, the signs of zn arising from the ambiguity of geometric mean iterations are chosen such that |wn − zn| ≤ |wn + zn| where wn and zn denote the individual arithmetic mean and geometric mean iterations of w and z, respectively. When |wn − zn| = |wn + zn|, there is an additional condition that Im<big>(</big><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" />⁠zn/wn⁠<big>)</big> > 0.<a href="#cite_note-21">[21]</a> Over the complex numbers, every elliptic curve has nine <a href="/wiki/Inflection_point" title="Inflection point">inflection points</a>. Every line through two of these points also passes through a third inflection point; the nine points and 12 lines formed in this way form a realization of the <a href="/wiki/Hesse_configuration" title="Hesse configuration">Hesse configuration</a>. <div class="mw-heading mw-heading2"><h2 id="The_dual_isogeny">The dual isogeny</h2>[<a href="/w/index.php?title=Elliptic_curve&action=edit&section=12" title="Edit section: The dual isogeny">edit</a>]</div> Given an <a href="/wiki/Isogeny" title="Isogeny">isogeny</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:E\to E'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>E</mi> <mo stretchy="false">→</mo> <msup> <mi>E</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:E\to E'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f71372f481489f2715ec157ae8ed865fc6b4a357" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.084ex; height:2.843ex;" alt="{\displaystyle f:E\to E'}" /></dd></dl> of elliptic curves of degree <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" />, the dual isogeny is an isogeny <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}:E'\to E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>:</mo> <msup> <mi>E</mi> <mo>′</mo> </msup> <mo stretchy="false">→</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}:E'\to E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e62f23bc992cc23388c99f462a3fdcf21d5ef2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.504ex; height:3.176ex;" alt="{\displaystyle {\hat {f}}:E'\to E}" /></dd></dl> of the same degree such that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ {\hat {f}}=[n].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∘</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\circ {\hat {f}}=[n].}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0902791fd7c1ef2cfbe31dfc7c8aadb95904acb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.606ex; height:3.343ex;" alt="{\displaystyle f\circ {\hat {f}}=[n].}" /></dd></dl> Here <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a26847bfc29bbeb4d6ef62ac3fd076378c0fd1db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.688ex; height:2.843ex;" alt="{\displaystyle [n]}" /> denotes the multiplication-by-<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /> isogeny <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e\mapsto ne}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo stretchy="false">↦</mo> <mi>n</mi> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e\mapsto ne}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f76d284539762b284267211b856f56112b5e460" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.176ex; height:1.843ex;" alt="{\displaystyle e\mapsto ne}" /> which has degree <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{2}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4846c73ba44c6ffdc37db7268c4f0d161b88dbe1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.096ex; height:2.676ex;" alt="{\displaystyle n^{2}.}" /> <div class="mw-heading mw-heading3"><h3 id="Construction_of_the_dual_isogeny">Construction of the dual isogeny</h3>[<a href="/w/index.php?title=Elliptic_curve&action=edit&section=13" title="Edit section: Construction of the dual isogeny">edit</a>]</div> Often only the existence of a dual isogeny is needed, but it can be explicitly given as the composition <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E'\to \operatorname {Div} ^{0}(E')\to \operatorname {Div} ^{0}(E)\to E,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>E</mi> <mo>′</mo> </msup> <mo stretchy="false">→</mo> <msup> <mi>Div</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>E</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <msup> <mi>Div</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>E</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E'\to \operatorname {Div} ^{0}(E')\to \operatorname {Div} ^{0}(E)\to E,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/527861f1e11edeb8483bc6987695c73d72d822c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.024ex; height:3.176ex;" alt="{\displaystyle E'\to \operatorname {Div} ^{0}(E')\to \operatorname {Div} ^{0}(E)\to E,}" /></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Div} ^{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>Div</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Div} ^{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0aa428478ae823d12870fde54c61c22c1d4f68cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.704ex; height:2.676ex;" alt="{\displaystyle \operatorname {Div} ^{0}}" /> is the group of <a href="/wiki/Divisor_(algebraic_geometry)" title="Divisor (algebraic geometry)">divisors</a> of degree 0. To do this, we need maps <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\to \operatorname {Div} ^{0}(E)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">→</mo> <msup> <mi>Div</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\to \operatorname {Div} ^{0}(E)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c908b46d36ffc17e10103516d6dd16599a4dc48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.679ex; height:3.176ex;" alt="{\displaystyle E\to \operatorname {Div} ^{0}(E)}" /> given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\to P-O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">→</mo> <mi>P</mi> <mo>−</mo> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\to P-O}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e424af4fcaad629bf3aa1fc671bed79330a9c8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.719ex; height:2.343ex;" alt="{\displaystyle P\to P-O}" /> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="{\displaystyle O}" /> is the neutral point of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Div} ^{0}(E)\to E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>Div</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Div} ^{0}(E)\to E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a731fde237ce2234a29333f3747350edf92e72af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.679ex; height:3.176ex;" alt="{\displaystyle \operatorname {Div} ^{0}(E)\to E}" /> given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum n_{P}P\to \sum n_{P}P.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∑</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mi>P</mi> <mo stretchy="false">→</mo> <mo>∑</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mi>P</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum n_{P}P\to \sum n_{P}P.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e320755ee85cb55cb90d7d0b3fa639927df4da91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:20.958ex; height:3.843ex;" alt="{\displaystyle \sum n_{P}P\to \sum n_{P}P.}" /> To see that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ {\hat {f}}=[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∘</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\circ {\hat {f}}=[n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b3abeb3d135c6c7ac30f660bc8eb29930af3d63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.959ex; height:3.343ex;" alt="{\displaystyle f\circ {\hat {f}}=[n]}" />, note that the original isogeny <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" /> can be written as a composite <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\to \operatorname {Div} ^{0}(E)\to \operatorname {Div} ^{0}(E')\to E',}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">→</mo> <msup> <mi>Div</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <msup> <mi>Div</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>E</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <msup> <mi>E</mi> <mo>′</mo> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\to \operatorname {Div} ^{0}(E)\to \operatorname {Div} ^{0}(E')\to E',}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccad8dd2914e8955bc8437b4e36649a3b7a64be0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.024ex; height:3.176ex;" alt="{\displaystyle E\to \operatorname {Div} ^{0}(E)\to \operatorname {Div} ^{0}(E')\to E',}" /></dd></dl> and that since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" /> is <a href="https://en.wiktionary.org/wiki/finite" class="extiw" title="wikt:finite">finite</a> of degree <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{*}f^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{*}f^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4977d2b1fe5112e127af10306049e8c7941d4288" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.568ex; height:2.676ex;" alt="{\displaystyle f_{*}f^{*}}" /> is multiplication by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Div} ^{0}(E').}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>Div</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>E</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Div} ^{0}(E').}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89c12c70843d27dad9367a21a71bb65ed5a95c41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.639ex; height:3.176ex;" alt="{\displaystyle \operatorname {Div} ^{0}(E').}" /> Alternatively, we can use the smaller <a href="/wiki/Picard_group" title="Picard group">Picard group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Pic} ^{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>Pic</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Pic} ^{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eda92267d30cfb06ea0bdcbd17ef1a2503ff7c47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.316ex; height:2.676ex;" alt="{\displaystyle \operatorname {Pic} ^{0}}" />, a <a href="/wiki/Factor_group" class="mw-redirect" title="Factor group">quotient</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Div} ^{0}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>Div</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Div} ^{0}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2217a96b1dcbf299a02c14b0b339e9cdb66730f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.351ex; height:2.676ex;" alt="{\displaystyle \operatorname {Div} ^{0}.}" /> The map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\to \operatorname {Div} ^{0}(E)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">→</mo> <msup> <mi>Div</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\to \operatorname {Div} ^{0}(E)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c908b46d36ffc17e10103516d6dd16599a4dc48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.679ex; height:3.176ex;" alt="{\displaystyle E\to \operatorname {Div} ^{0}(E)}" /> descends to an <a href="/wiki/Isomorphism" title="Isomorphism">isomorphism</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\to \operatorname {Pic} ^{0}(E).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">→</mo> <msup> <mi>Pic</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\to \operatorname {Pic} ^{0}(E).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2745c8164f2eac9a591348a89d49ba804c2b08d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.938ex; height:3.176ex;" alt="{\displaystyle E\to \operatorname {Pic} ^{0}(E).}" /> The dual isogeny is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E'\to \operatorname {Pic} ^{0}(E')\to \operatorname {Pic} ^{0}(E)\to E.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>E</mi> <mo>′</mo> </msup> <mo stretchy="false">→</mo> <msup> <mi>Pic</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>E</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <msup> <mi>Pic</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>E</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E'\to \operatorname {Pic} ^{0}(E')\to \operatorname {Pic} ^{0}(E)\to E.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9923189eccfcc871e92fc2e75dc6c336fd4ace8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.249ex; height:3.176ex;" alt="{\displaystyle E'\to \operatorname {Pic} ^{0}(E')\to \operatorname {Pic} ^{0}(E)\to E.}" /></dd></dl> Note that the relation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ {\hat {f}}=[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∘</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\circ {\hat {f}}=[n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b3abeb3d135c6c7ac30f660bc8eb29930af3d63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.959ex; height:3.343ex;" alt="{\displaystyle f\circ {\hat {f}}=[n]}" /> also implies the conjugate relation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}\circ f=[n].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>∘</mo> <mi>f</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}\circ f=[n].}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db9b4cff65c3c9027f5bc75a4aefebe3110726d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.606ex; height:3.343ex;" alt="{\displaystyle {\hat {f}}\circ f=[n].}" /> Indeed, let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi ={\hat {f}}\circ f.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>∘</mo> <mi>f</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi ={\hat {f}}\circ f.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/903b57001629f00d8e8f2ad69874a7cf2229cb1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.303ex; height:3.176ex;" alt="{\displaystyle \phi ={\hat {f}}\circ f.}" /> Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi \circ {\hat {f}}={\hat {f}}\circ [n]=[n]\circ {\hat {f}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>∘</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>∘</mo> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>∘</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi \circ {\hat {f}}={\hat {f}}\circ [n]=[n]\circ {\hat {f}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82b6080d67f899f96c9bfaae4f531c7df81a957e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.287ex; height:3.343ex;" alt="{\displaystyle \phi \circ {\hat {f}}={\hat {f}}\circ [n]=[n]\circ {\hat {f}}.}" /> But <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14ce989fd75da938ec6f95a0cdb71037b23a11cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.699ex; height:3.176ex;" alt="{\displaystyle {\hat {f}}}" /> is <a href="/wiki/Surjection" class="mw-redirect" title="Surjection">surjective</a>, so we must have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi =[n].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi =[n].}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9e3f6a2c5064d9fd024a8b2c6609d2638012b10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.819ex; height:2.843ex;" alt="{\displaystyle \phi =[n].}" /> <div class="mw-heading mw-heading2"><h2 id="Algorithms_that_use_elliptic_curves">Algorithms that use elliptic curves</h2>[<a href="/w/index.php?title=Elliptic_curve&action=edit&section=14" title="Edit section: Algorithms that use elliptic curves">edit</a>]</div> Elliptic curves over finite fields are used in some <a href="/wiki/Cryptography" title="Cryptography">cryptographic</a> applications as well as for <a href="/wiki/Integer_factorization" title="Integer factorization">integer factorization</a>. Typically, the general idea in these applications is that a known <a href="/wiki/Algorithm" title="Algorithm">algorithm</a> which makes use of certain finite groups is rewritten to use the groups of rational points of elliptic curves. For more see also: <ul><li><a href="/wiki/Elliptic_curve_cryptography" class="mw-redirect" title="Elliptic curve cryptography">Elliptic curve cryptography</a> <ul><li><a href="/wiki/Elliptic-curve_Diffie%E2%80%93Hellman" title="Elliptic-curve Diffie–Hellman">Elliptic-curve Diffie–Hellman</a> key exchange (ECDH)</li> <li><a href="/wiki/Supersingular_isogeny_key_exchange" title="Supersingular isogeny key exchange">Supersingular isogeny key exchange</a></li> <li><a href="/wiki/Elliptic_Curve_Digital_Signature_Algorithm" title="Elliptic Curve Digital Signature Algorithm">Elliptic curve digital signature algorithm</a> (ECDSA)</li> <li><a href="/wiki/EdDSA" title="EdDSA">EdDSA</a> digital signature algorithm</li> <li><a href="/wiki/Dual_EC_DRBG" title="Dual EC DRBG">Dual EC DRBG</a> random number generator</li></ul></li> <li><a href="/wiki/Lenstra_elliptic-curve_factorization" title="Lenstra elliptic-curve factorization">Lenstra elliptic-curve factorization</a></li> <li><a href="/wiki/Elliptic_curve_primality_proving" class="mw-redirect" title="Elliptic curve primality proving">Elliptic curve primality proving</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Alternative_representations_of_elliptic_curves">Alternative representations of elliptic curves</h2>[<a href="/w/index.php?title=Elliptic_curve&action=edit&section=15" title="Edit section: Alternative representations of elliptic curves">edit</a>]</div> <ul><li><a href="/wiki/Hessian_form_of_an_elliptic_curve" title="Hessian form of an elliptic curve">Hessian curve</a></li> <li><a href="/wiki/Edwards_curve" title="Edwards curve">Edwards curve</a></li> <li><a href="/wiki/Twists_of_curves" class="mw-redirect" title="Twists of curves">Twisted curve</a></li> <li><a href="/wiki/Twisted_Hessian_curves" title="Twisted Hessian curves">Twisted Hessian curve</a></li> <li><a href="/wiki/Twisted_Edwards_curve" title="Twisted Edwards curve">Twisted Edwards curve</a></li> <li><a href="/wiki/Doubling-oriented_Doche%E2%80%93Icart%E2%80%93Kohel_curve" title="Doubling-oriented Doche–Icart–Kohel curve">Doubling-oriented Doche–Icart–Kohel curve</a></li> <li><a href="/wiki/Tripling-oriented_Doche%E2%80%93Icart%E2%80%93Kohel_curve" title="Tripling-oriented Doche–Icart–Kohel curve">Tripling-oriented Doche–Icart–Kohel curve</a></li> <li><a href="/wiki/Jacobian_curve" title="Jacobian curve">Jacobian curve</a></li> <li><a href="/wiki/Montgomery_curve" title="Montgomery curve">Montgomery curve</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Elliptic_curve&action=edit&section=16" title="Edit section: See also">edit</a>]</div> <ul><li><a href="/wiki/Arithmetic_dynamics" title="Arithmetic dynamics">Arithmetic dynamics</a></li> <li><a href="/wiki/Elliptic_algebra" title="Elliptic algebra">Elliptic algebra</a></li> <li><a href="/wiki/Elliptic_surface" title="Elliptic surface">Elliptic surface</a></li> <li><a href="/wiki/Comparison_of_computer_algebra_systems" class="mw-redirect" title="Comparison of computer algebra systems">Comparison of computer algebra systems</a></li> <li><a href="/wiki/Isogeny" title="Isogeny">Isogeny</a></li> <li><a href="/wiki/J-line" title="J-line">j-line</a></li> <li><a href="/wiki/Level_structure_(algebraic_geometry)" title="Level structure (algebraic geometry)">Level structure (algebraic geometry)</a></li> <li><a href="/wiki/Modularity_theorem" title="Modularity theorem">Modularity theorem</a></li> <li><a href="/wiki/Moduli_stack_of_elliptic_curves" title="Moduli stack of elliptic curves">Moduli stack of elliptic curves</a></li> <li><a href="/wiki/Nagell%E2%80%93Lutz_theorem" title="Nagell–Lutz theorem">Nagell–Lutz theorem</a></li> <li><a href="/wiki/Riemann%E2%80%93Hurwitz_formula" title="Riemann–Hurwitz formula">Riemann–Hurwitz formula</a></li> <li><a href="/wiki/Wiles%27s_proof_of_Fermat%27s_Last_Theorem" title="Wiles's proof of Fermat's Last Theorem">Wiles's proof of Fermat's Last Theorem</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Elliptic_curve&action=edit&section=17" title="Edit section: Notes">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFSarli2012" class="citation journal cs1">Sarli, J. (2012). "Conics in the hyperbolic plane intrinsic to the collineation group". J. Geom. 103: 131–148. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00022-012-0115-5">10.1007/s00022-012-0115-5</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119588289">119588289</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=J.+Geom.&rft.atitle=Conics+in+the+hyperbolic+plane+intrinsic+to+the+collineation+group&rft.volume=103&rft.pages=%3Cspan+class%3D%22nowrap%22%3E131-%3C%2Fspan%3E148&rft.date=2012&rft_id=info%3Adoi%2F10.1007%2Fs00022-012-0115-5&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119588289%23id-name%3DS2CID&rft.aulast=Sarli&rft.aufirst=J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+curve" class="Z3988"> </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> Silverman <a href="#CITEREFSilverman1986">1986</a>, III.1 Weierstrass Equations (p.45) </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> T. Nagell, L'analyse indéterminée de degré supérieur, Mémorial des sciences mathématiques 39, Paris, Gauthier-Villars, 1929, pp. 56–59. </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> OEIS: <a rel="nofollow" class="external free" href="https://oeis.org/A029728">https://oeis.org/A029728</a> </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSiksek1995" class="citation cs2">Siksek, Samir (1995), Descents on Curves of Genus 1 (Ph.D. thesis), University of Exeter, pp. 16–17, <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/10871%2F8323">10871/8323</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Descents+on+Curves+of+Genus+1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E16-%3C%2Fspan%3E17&rft.pub=University+of+Exeter&rft.date=1995&rft_id=info%3Ahdl%2F10871%2F8323&rft.aulast=Siksek&rft.aufirst=Samir&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+curve" class="Z3988"> </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> Silverman <a href="#CITEREFSilverman1986">1986</a>, Theorem 4.1 </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> Silverman <a href="#CITEREFSilverman1986">1986</a>, pp. 199–205 </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> See also <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCassels1986" class="citation journal cs1"><a href="/wiki/J._W._S._Cassels" title="J. W. S. Cassels">Cassels, J. W. S.</a> (1986). "Mordell's Finite Basis Theorem Revisited". <a href="/wiki/Mathematical_Proceedings_of_the_Cambridge_Philosophical_Society" title="Mathematical Proceedings of the Cambridge Philosophical Society">Mathematical Proceedings of the Cambridge Philosophical Society</a>. 100 (1): 31–41. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1986MPCPS.100...31C">1986MPCPS.100...31C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0305004100065841">10.1017/S0305004100065841</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematical+Proceedings+of+the+Cambridge+Philosophical+Society&rft.atitle=Mordell%27s+Finite+Basis+Theorem+Revisited&rft.volume=100&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E31-%3C%2Fspan%3E41&rft.date=1986&rft_id=info%3Adoi%2F10.1017%2FS0305004100065841&rft_id=info%3Abibcode%2F1986MPCPS.100...31C&rft.aulast=Cassels&rft.aufirst=J.+W.+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+curve" class="Z3988"> and the comment of A. Weil on the genesis of his work: A. Weil, Collected Papers, vol. 1, 520–521. </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDujella" class="citation web cs1"><a href="/wiki/Andrej_Dujella" title="Andrej Dujella">Dujella, Andrej</a>. <a rel="nofollow" class="external text" href="http://web.math.pmf.unizg.hr/~duje/tors/rankhist.html">"History of elliptic curves rank records"</a>. University of Zagreb.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=History+of+elliptic+curves+rank+records&rft.pub=University+of+Zagreb&rft.aulast=Dujella&rft.aufirst=Andrej&rft_id=http%3A%2F%2Fweb.math.pmf.unizg.hr%2F~duje%2Ftors%2Frankhist.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+curve" class="Z3988"> </li> <li id="cite_note-10"><a href="#cite_ref-10">^</a> Silverman <a href="#CITEREFSilverman1986">1986</a>, Theorem 7.5 </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> Silverman <a href="#CITEREFSilverman1986">1986</a>, Remark 7.8 in Ch. VIII </li> <li id="cite_note-12"><a href="#cite_ref-12">^</a> The definition is formal, the exponential of this <a href="/wiki/Power_series" title="Power series">power series</a> without constant term denotes the usual development. </li> <li id="cite_note-13"><a href="#cite_ref-13">^</a> see for example <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSilverman2006" class="citation web cs1">Silverman, Joseph H. (2006). <a rel="nofollow" class="external text" href="https://www.math.brown.edu/~jhs/Presentations/WyomingEllipticCurve.pdf">"An Introduction to the Theory of Elliptic Curves"</a> (PDF). Summer School on Computational Number Theory and Applications to Cryptography. University of Wyoming.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Summer+School+on+Computational+Number+Theory+and+Applications+to+Cryptography&rft.atitle=An+Introduction+to+the+Theory+of+Elliptic+Curves&rft.date=2006&rft.aulast=Silverman&rft.aufirst=Joseph+H.&rft_id=https%3A%2F%2Fwww.math.brown.edu%2F~jhs%2FPresentations%2FWyomingEllipticCurve.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+curve" class="Z3988"> </li> <li id="cite_note-14"><a href="#cite_ref-14">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.lmfdb.org/knowledge/show/ec.bad_reduction">"LMFDB - Bad reduction of an elliptic curve at a prime (Reviewed)"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=LMFDB+-+Bad+reduction+of+an+elliptic+curve+at+a+prime+%28Reviewed%29&rft_id=https%3A%2F%2Fwww.lmfdb.org%2Fknowledge%2Fshow%2Fec.bad_reduction&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+curve" class="Z3988"> </li> <li id="cite_note-15"><a href="#cite_ref-15">^</a> Koblitz <a href="#CITEREFKoblitz1993">1993</a> </li> <li id="cite_note-16"><a href="#cite_ref-16">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHeath-Brown2004" class="citation journal cs1">Heath-Brown, D. R. (2004). "The Average Analytic Rank of Elliptic Curves". Duke Mathematical Journal. 122 (3): 591–623. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0305114">math/0305114</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1215%2FS0012-7094-04-12235-3">10.1215/S0012-7094-04-12235-3</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:15216987">15216987</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Duke+Mathematical+Journal&rft.atitle=The+Average+Analytic+Rank+of+Elliptic+Curves&rft.volume=122&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E591-%3C%2Fspan%3E623&rft.date=2004&rft_id=info%3Aarxiv%2Fmath%2F0305114&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A15216987%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1215%2FS0012-7094-04-12235-3&rft.aulast=Heath-Brown&rft.aufirst=D.+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+curve" class="Z3988"> </li> <li id="cite_note-17"><a href="#cite_ref-17">^</a> See Koblitz <a href="#CITEREFKoblitz1994">1994</a>, p. 158 </li> <li id="cite_note-18"><a href="#cite_ref-18">^</a> Koblitz <a href="#CITEREFKoblitz1994">1994</a>, p. 160 </li> <li id="cite_note-19"><a href="#cite_ref-19">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHarrisShepherd-BarronTaylor2010" class="citation journal cs1">Harris, M.; Shepherd-Barron, N.; Taylor, R. (2010). <a rel="nofollow" class="external text" href="https://doi.org/10.4007%2Fannals.2010.171.779">"A family of Calabi–Yau varieties and potential automorphy"</a>. <a href="/wiki/Annals_of_Mathematics" title="Annals of Mathematics">Annals of Mathematics</a>. 171 (2): 779–813. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4007%2Fannals.2010.171.779">10.4007/annals.2010.171.779</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematics&rft.atitle=A+family+of+Calabi%E2%80%93Yau+varieties+and+potential+automorphy&rft.volume=171&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E779-%3C%2Fspan%3E813&rft.date=2010&rft_id=info%3Adoi%2F10.4007%2Fannals.2010.171.779&rft.aulast=Harris&rft.aufirst=M.&rft.au=Shepherd-Barron%2C+N.&rft.au=Taylor%2C+R.&rft_id=https%3A%2F%2Fdoi.org%2F10.4007%252Fannals.2010.171.779&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+curve" class="Z3988"> </li> <li id="cite_note-20"><a href="#cite_ref-20">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMerel1996" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Lo%C3%AFc_Merel" title="Loïc Merel">Merel, L.</a> (1996). "Bornes pour la torsion des courbes elliptiques sur les corps de nombres". <a href="/wiki/Inventiones_Mathematicae" title="Inventiones Mathematicae">Inventiones Mathematicae</a> (in French). 124 (1–3): 437–449. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1996InMat.124..437M">1996InMat.124..437M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs002220050059">10.1007/s002220050059</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:3590991">3590991</a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0936.11037">0936.11037</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Inventiones+Mathematicae&rft.atitle=Bornes+pour+la+torsion+des+courbes+elliptiques+sur+les+corps+de+nombres&rft.volume=124&rft.issue=%3Cspan+class%3D%22nowrap%22%3E1%E2%80%93%3C%2Fspan%3E3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E437-%3C%2Fspan%3E449&rft.date=1996&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0936.11037%23id-name%3DZbl&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A3590991%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs002220050059&rft_id=info%3Abibcode%2F1996InMat.124..437M&rft.aulast=Merel&rft.aufirst=L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+curve" class="Z3988"> </li> <li id="cite_note-21"><a href="#cite_ref-21">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWing_Tat_Chow2018" class="citation web cs1">Wing Tat Chow, Rudolf (2018). <a rel="nofollow" class="external text" href="http://etheses.whiterose.ac.uk/20887/1/Final.pdf">"The Arithmetic-Geometric Mean and Periods of Curves of Genus 1 and 2"</a> (PDF). White Rose eTheses Online. p. 12.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=White+Rose+eTheses+Online&rft.atitle=The+Arithmetic-Geometric+Mean+and+Periods+of+Curves+of+Genus+1+and+2&rft.pages=12&rft.date=2018&rft.aulast=Wing+Tat+Chow&rft.aufirst=Rudolf&rft_id=http%3A%2F%2Fetheses.whiterose.ac.uk%2F20887%2F1%2FFinal.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+curve" class="Z3988"> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Elliptic_curve&action=edit&section=18" title="Edit section: References">edit</a>]</div> <a href="/wiki/Serge_Lang" title="Serge Lang">Serge Lang</a>, in the introduction to the book cited below, stated that "It is possible to write endlessly on elliptic curves. (This is not a threat.)" The following short list is thus at best a guide to the vast expository literature available on the theoretical, algorithmic, and cryptographic aspects of elliptic curves. <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFIan_BlakeGadiel_SeroussiNigel_Smart2000" class="citation book cs1">Ian Blake; Gadiel Seroussi; <a href="/wiki/Nigel_Smart_(cryptographer)" title="Nigel Smart (cryptographer)">Nigel Smart</a> (2000). Elliptic Curves in Cryptography. LMS Lecture Notes. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-65374-6" title="Special:BookSources/0-521-65374-6"><bdi>0-521-65374-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elliptic+Curves+in+Cryptography&rft.series=LMS+Lecture+Notes&rft.pub=Cambridge+University+Press&rft.date=2000&rft.isbn=0-521-65374-6&rft.au=Ian+Blake&rft.au=Gadiel+Seroussi&rft.au=Nigel+Smart&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+curve" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBrown2000" class="citation journal cs1"><a href="/wiki/Ezra_Brown" title="Ezra Brown">Brown, Ezra</a> (2000). "Three Fermat Trails to Elliptic Curves". The College Mathematics Journal. 31 (3): 162–172. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F07468342.2000.11974137">10.1080/07468342.2000.11974137</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:5591395">5591395</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+College+Mathematics+Journal&rft.atitle=Three+Fermat+Trails+to+Elliptic+Curves&rft.volume=31&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E162-%3C%2Fspan%3E172&rft.date=2000&rft_id=info%3Adoi%2F10.1080%2F07468342.2000.11974137&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A5591395%23id-name%3DS2CID&rft.aulast=Brown&rft.aufirst=Ezra&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+curve" class="Z3988">, winner of the MAA writing prize the <a href="/wiki/George_P%C3%B3lya_Award" title="George Pólya Award">George Pólya Award</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRichard_CrandallCarl_Pomerance2001" class="citation book cs1"><a href="/wiki/Richard_Crandall" title="Richard Crandall">Richard Crandall</a>; <a href="/wiki/Carl_Pomerance" title="Carl Pomerance">Carl Pomerance</a> (2001). "Chapter 7: Elliptic Curve Arithmetic". Prime Numbers: A Computational Perspective (1st ed.). Springer-Verlag. pp. 285–352. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-94777-9" title="Special:BookSources/0-387-94777-9"><bdi>0-387-94777-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+7%3A+Elliptic+Curve+Arithmetic&rft.btitle=Prime+Numbers%3A+A+Computational+Perspective&rft.pages=%3Cspan+class%3D%22nowrap%22%3E285-%3C%2Fspan%3E352&rft.edition=1st&rft.pub=Springer-Verlag&rft.date=2001&rft.isbn=0-387-94777-9&rft.au=Richard+Crandall&rft.au=Carl+Pomerance&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+curve" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCremona1997" class="citation book cs1">Cremona, John (1997). <a rel="nofollow" class="external text" href="http://www.warwick.ac.uk/staff/J.E.Cremona//book/fulltext/index.html">Algorithms for Modular Elliptic Curves</a> (2nd ed.). Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-59820-6" title="Special:BookSources/0-521-59820-6"><bdi>0-521-59820-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algorithms+for+Modular+Elliptic+Curves&rft.edition=2nd&rft.pub=Cambridge+University+Press&rft.date=1997&rft.isbn=0-521-59820-6&rft.aulast=Cremona&rft.aufirst=John&rft_id=http%3A%2F%2Fwww.warwick.ac.uk%2Fstaff%2FJ.E.Cremona%2F%2Fbook%2Ffulltext%2Findex.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+curve" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDarrel_Hankerson,_Alfred_Menezes_and_Scott_Vanstone2004" class="citation book cs1">Darrel Hankerson, <a href="/wiki/Alfred_Menezes" title="Alfred Menezes">Alfred Menezes</a> and <a href="/wiki/Scott_Vanstone" title="Scott Vanstone">Scott Vanstone</a> (2004). <a rel="nofollow" class="external text" href="http://www.cacr.math.uwaterloo.ca/ecc/">Guide to Elliptic Curve Cryptography</a>. <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-95273-X" title="Special:BookSources/0-387-95273-X"><bdi>0-387-95273-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Guide+to+Elliptic+Curve+Cryptography&rft.pub=Springer&rft.date=2004&rft.isbn=0-387-95273-X&rft.au=Darrel+Hankerson%2C+Alfred+Menezes+and+Scott+Vanstone&rft_id=http%3A%2F%2Fwww.cacr.math.uwaterloo.ca%2Fecc%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+curve" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHardyWright2008" class="citation book cs1"><a href="/wiki/G._H._Hardy" title="G. H. Hardy">Hardy, G. H.</a>; <a href="/wiki/E._M._Wright" title="E. M. Wright">Wright, E. M.</a> (2008) [1938]. An Introduction to the Theory of Numbers. Revised by <a href="/wiki/Roger_Heath-Brown" title="Roger Heath-Brown">D. R. Heath-Brown</a> and <a href="/wiki/Joseph_H._Silverman" title="Joseph H. Silverman">J. H. Silverman</a>. Foreword by <a href="/wiki/Andrew_Wiles" title="Andrew Wiles">Andrew Wiles</a>. (6th ed.). Oxford: <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-921986-5" title="Special:BookSources/978-0-19-921986-5"><bdi>978-0-19-921986-5</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2445243">2445243</a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1159.11001">1159.11001</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+the+Theory+of+Numbers&rft.place=Oxford&rft.edition=6th&rft.pub=Oxford+University+Press&rft.date=2008&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1159.11001%23id-name%3DZbl&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2445243%23id-name%3DMR&rft.isbn=978-0-19-921986-5&rft.aulast=Hardy&rft.aufirst=G.+H.&rft.au=Wright%2C+E.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+curve" class="Z3988"> Chapter XXV</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHellegouarch2001" class="citation book cs1">Hellegouarch, Yves (2001). Invitation aux mathématiques de Fermat-Wiles. Paris: Dunod. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-2-10-005508-1" title="Special:BookSources/978-2-10-005508-1"><bdi>978-2-10-005508-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Invitation+aux+math%C3%A9matiques+de+Fermat-Wiles&rft.place=Paris&rft.pub=Dunod&rft.date=2001&rft.isbn=978-2-10-005508-1&rft.aulast=Hellegouarch&rft.aufirst=Yves&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+curve" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHusemöller2004" class="citation book cs1"><a href="/wiki/Dale_Husemoller" title="Dale Husemoller">Husemöller, Dale</a> (2004). Elliptic Curves. <a href="/wiki/Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a>. Vol. 111 (2nd ed.). Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-95490-2" title="Special:BookSources/0-387-95490-2"><bdi>0-387-95490-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elliptic+Curves&rft.series=Graduate+Texts+in+Mathematics&rft.edition=2nd&rft.pub=Springer&rft.date=2004&rft.isbn=0-387-95490-2&rft.aulast=Husem%C3%B6ller&rft.aufirst=Dale&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+curve" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKenneth_IrelandMichael_I._Rosen1998" class="citation book cs1">Kenneth Ireland; <a href="/wiki/Michael_Rosen_(mathematician)" title="Michael Rosen (mathematician)">Michael I. Rosen</a> (1998). "Chapters 18 and 19". A Classical Introduction to Modern Number Theory. Graduate Texts in Mathematics. Vol. 84 (2nd revised ed.). Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-97329-X" title="Special:BookSources/0-387-97329-X"><bdi>0-387-97329-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapters+18+and+19&rft.btitle=A+Classical+Introduction+to+Modern+Number+Theory&rft.series=Graduate+Texts+in+Mathematics&rft.edition=2nd+revised&rft.pub=Springer&rft.date=1998&rft.isbn=0-387-97329-X&rft.au=Kenneth+Ireland&rft.au=Michael+I.+Rosen&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+curve" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKnapp2018" class="citation book cs1"><a href="/wiki/Anthony_W._Knapp" title="Anthony W. Knapp">Knapp, Anthony W.</a> (2018) [1992]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Xf9ZDwAAQBAJ">Elliptic Curves</a>. Mathematical Notes. Vol. 40. Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780691186900" title="Special:BookSources/9780691186900"><bdi>9780691186900</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elliptic+Curves&rft.series=Mathematical+Notes&rft.pub=Princeton+University+Press&rft.date=2018&rft.isbn=9780691186900&rft.aulast=Knapp&rft.aufirst=Anthony+W.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DXf9ZDwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+curve" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKoblitz1993" class="citation book cs1"><a href="/wiki/Neal_Koblitz" title="Neal Koblitz">Koblitz, Neal</a> (1993). Introduction to Elliptic Curves and Modular Forms. Graduate Texts in Mathematics. Vol. 97 (2nd ed.). Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-97966-2" title="Special:BookSources/0-387-97966-2"><bdi>0-387-97966-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Elliptic+Curves+and+Modular+Forms&rft.series=Graduate+Texts+in+Mathematics&rft.edition=2nd&rft.pub=Springer-Verlag&rft.date=1993&rft.isbn=0-387-97966-2&rft.aulast=Koblitz&rft.aufirst=Neal&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+curve" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKoblitz1994" class="citation book cs1"><a href="/wiki/Neal_Koblitz" title="Neal Koblitz">Koblitz, Neal</a> (1994). "Chapter 6". A Course in Number Theory and Cryptography. Graduate Texts in Mathematics. Vol. 114 (2nd ed.). Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-94293-9" title="Special:BookSources/0-387-94293-9"><bdi>0-387-94293-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+6&rft.btitle=A+Course+in+Number+Theory+and+Cryptography&rft.series=Graduate+Texts+in+Mathematics&rft.edition=2nd&rft.pub=Springer-Verlag&rft.date=1994&rft.isbn=0-387-94293-9&rft.aulast=Koblitz&rft.aufirst=Neal&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+curve" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSerge_Lang1978" class="citation book cs1"><a href="/wiki/Serge_Lang" title="Serge Lang">Serge Lang</a> (1978). Elliptic curves: Diophantine analysis. Grundlehren der mathematischen Wissenschaften. Vol. 231. Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-08489-4" title="Special:BookSources/3-540-08489-4"><bdi>3-540-08489-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elliptic+curves%3A+Diophantine+analysis&rft.series=Grundlehren+der+mathematischen+Wissenschaften&rft.pub=Springer-Verlag&rft.date=1978&rft.isbn=3-540-08489-4&rft.au=Serge+Lang&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+curve" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHenry_McKeanVictor_Moll1999" class="citation book cs1"><a href="/wiki/Henry_McKean" title="Henry McKean">Henry McKean</a>; <a href="/wiki/Victor_Hugo_Moll" class="mw-redirect" title="Victor Hugo Moll">Victor Moll</a> (1999). Elliptic curves: function theory, geometry and arithmetic. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-65817-9" title="Special:BookSources/0-521-65817-9"><bdi>0-521-65817-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elliptic+curves%3A+function+theory%2C+geometry+and+arithmetic&rft.pub=Cambridge+University+Press&rft.date=1999&rft.isbn=0-521-65817-9&rft.au=Henry+McKean&rft.au=Victor+Moll&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+curve" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFIvan_NivenHerbert_S._ZuckermanHugh_Montgomery1991" class="citation book cs1"><a href="/wiki/Ivan_M._Niven" title="Ivan M. Niven">Ivan Niven</a>; Herbert S. Zuckerman; <a href="/wiki/Hugh_Montgomery_(mathematician)" class="mw-redirect" title="Hugh Montgomery (mathematician)">Hugh Montgomery</a> (1991). <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoth0000nive">"Section 5.7"</a>. An introduction to the theory of numbers (5th ed.). John Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-54600-3" title="Special:BookSources/0-471-54600-3"><bdi>0-471-54600-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Section+5.7&rft.btitle=An+introduction+to+the+theory+of+numbers&rft.edition=5th&rft.pub=John+Wiley&rft.date=1991&rft.isbn=0-471-54600-3&rft.au=Ivan+Niven&rft.au=Herbert+S.+Zuckerman&rft.au=Hugh+Montgomery&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontoth0000nive&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+curve" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSilverman1986" class="citation book cs1"><a href="/wiki/Joseph_H._Silverman" title="Joseph H. Silverman">Silverman, Joseph H.</a> (1986). The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics. Vol. 106. Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-96203-4" title="Special:BookSources/0-387-96203-4"><bdi>0-387-96203-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Arithmetic+of+Elliptic+Curves&rft.series=Graduate+Texts+in+Mathematics&rft.pub=Springer-Verlag&rft.date=1986&rft.isbn=0-387-96203-4&rft.aulast=Silverman&rft.aufirst=Joseph+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+curve" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFJoseph_H._Silverman1994" class="citation book cs1"><a href="/wiki/Joseph_H._Silverman" title="Joseph H. Silverman">Joseph H. Silverman</a> (1994). Advanced Topics in the Arithmetic of Elliptic Curves. Graduate Texts in Mathematics. Vol. 151. Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-94328-5" title="Special:BookSources/0-387-94328-5"><bdi>0-387-94328-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Advanced+Topics+in+the+Arithmetic+of+Elliptic+Curves&rft.series=Graduate+Texts+in+Mathematics&rft.pub=Springer-Verlag&rft.date=1994&rft.isbn=0-387-94328-5&rft.au=Joseph+H.+Silverman&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+curve" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFJoseph_H._SilvermanJohn_Tate1992" class="citation book cs1"><a href="/wiki/Joseph_H._Silverman" title="Joseph H. Silverman">Joseph H. Silverman</a>; <a href="/wiki/John_Tate_(mathematician)" title="John Tate (mathematician)">John Tate</a> (1992). Rational Points on Elliptic Curves. Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-97825-9" title="Special:BookSources/0-387-97825-9"><bdi>0-387-97825-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Rational+Points+on+Elliptic+Curves&rft.pub=Springer-Verlag&rft.date=1992&rft.isbn=0-387-97825-9&rft.au=Joseph+H.+Silverman&rft.au=John+Tate&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+curve" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFJohn_Tate1974" class="citation journal cs1"><a href="/wiki/John_Tate_(mathematician)" title="John Tate (mathematician)">John Tate</a> (1974). "The arithmetic of elliptic curves". <a href="/wiki/Inventiones_Mathematicae" title="Inventiones Mathematicae">Inventiones Mathematicae</a>. 23 (3–4): 179–206. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1974InMat..23..179T">1974InMat..23..179T</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01389745">10.1007/BF01389745</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120008651">120008651</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Inventiones+Mathematicae&rft.atitle=The+arithmetic+of+elliptic+curves&rft.volume=23&rft.issue=%3Cspan+class%3D%22nowrap%22%3E3%E2%80%93%3C%2Fspan%3E4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E179-%3C%2Fspan%3E206&rft.date=1974&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120008651%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF01389745&rft_id=info%3Abibcode%2F1974InMat..23..179T&rft.au=John+Tate&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+curve" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLawrence_Washington2003" class="citation book cs1"><a href="/wiki/Lawrence_C._Washington" title="Lawrence C. Washington">Lawrence Washington</a> (2003). <a rel="nofollow" class="external text" href="https://archive.org/details/ellipticcurvesnu0000wash">Elliptic Curves: Number Theory and Cryptography</a>. Chapman & Hall/CRC. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-58488-365-0" title="Special:BookSources/1-58488-365-0"><bdi>1-58488-365-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elliptic+Curves%3A+Number+Theory+and+Cryptography&rft.pub=Chapman+%26+Hall%2FCRC&rft.date=2003&rft.isbn=1-58488-365-0&rft.au=Lawrence+Washington&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fellipticcurvesnu0000wash&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+curve" class="Z3988"></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Elliptic_curve&action=edit&section=19" title="Edit section: External links">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid 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Mathematics</a>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Elliptic+curve&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DElliptic_curve&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+curve" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. 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href="/wiki/Template_talk:Algebraic_curves_navbox" title="Template talk:Algebraic curves navbox"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Algebraic_curves_navbox" title="Special:EditPage/Template:Algebraic curves navbox"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Topics_in_algebraic_curves148" style="font-size:114%;margin:0 4em">Topics in <a href="/wiki/Algebraic_curve" title="Algebraic curve">algebraic curves</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Rational_curve" class="mw-redirect" title="Rational curve">Rational curves</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Five_points_determine_a_conic" title="Five points determine a conic">Five points determine a conic</a></li> <li><a href="/wiki/Projective_line" title="Projective line">Projective line</a></li> <li><a href="/wiki/Rational_normal_curve" title="Rational normal curve">Rational normal curve</a></li> <li><a href="/wiki/Riemann_sphere" title="Riemann sphere">Riemann sphere</a></li> <li><a href="/wiki/Twisted_cubic" title="Twisted cubic">Twisted cubic</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a class="mw-selflink selflink">Elliptic curves</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Analytic theory</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Elliptic_function" title="Elliptic function">Elliptic function</a></li> <li><a href="/wiki/Elliptic_integral" title="Elliptic integral">Elliptic integral</a></li> <li><a href="/wiki/Fundamental_pair_of_periods" title="Fundamental pair of periods">Fundamental pair of periods</a></li> <li><a href="/wiki/Modular_form" title="Modular form">Modular form</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Arithmetic theory</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Counting_points_on_elliptic_curves" title="Counting points on elliptic curves">Counting points on elliptic curves</a></li> <li><a href="/wiki/Division_polynomials" title="Division polynomials">Division polynomials</a></li> <li><a href="/wiki/Hasse%27s_theorem_on_elliptic_curves" title="Hasse's theorem on elliptic curves">Hasse's theorem on elliptic curves</a></li> <li><a href="/wiki/Mazur%27s_torsion_theorem" class="mw-redirect" title="Mazur's torsion theorem">Mazur's torsion theorem</a></li> <li><a href="/wiki/Modular_elliptic_curve" title="Modular elliptic curve">Modular elliptic curve</a></li> <li><a href="/wiki/Modularity_theorem" title="Modularity theorem">Modularity theorem</a></li> <li><a href="/wiki/Mordell%E2%80%93Weil_theorem" title="Mordell–Weil theorem">Mordell–Weil theorem</a></li> <li><a href="/wiki/Nagell%E2%80%93Lutz_theorem" title="Nagell–Lutz theorem">Nagell–Lutz theorem</a></li> <li><a href="/wiki/Supersingular_elliptic_curve" title="Supersingular elliptic curve">Supersingular elliptic curve</a></li> <li><a href="/wiki/Schoof%27s_algorithm" title="Schoof's algorithm">Schoof's algorithm</a></li> <li><a href="/wiki/Schoof%E2%80%93Elkies%E2%80%93Atkin_algorithm" title="Schoof–Elkies–Atkin algorithm">Schoof–Elkies–Atkin algorithm</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Elliptic_curve_cryptography" class="mw-redirect" title="Elliptic curve cryptography">Elliptic curve cryptography</a></li> <li><a href="/wiki/Elliptic_curve_primality" title="Elliptic curve primality">Elliptic curve primality</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Higher genus</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/De_Franchis_theorem" title="De Franchis theorem">De Franchis theorem</a></li> <li><a href="/wiki/Faltings%27s_theorem" title="Faltings's theorem">Faltings's theorem</a></li> <li><a href="/wiki/Hurwitz%27s_automorphisms_theorem" title="Hurwitz's automorphisms theorem">Hurwitz's automorphisms theorem</a></li> <li><a href="/wiki/Hurwitz_surface" title="Hurwitz surface">Hurwitz surface</a></li> <li><a href="/wiki/Hyperelliptic_curve" title="Hyperelliptic curve">Hyperelliptic curve</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Plane_curve" title="Plane curve">Plane curves</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/AF%2BBG_theorem" title="AF+BG theorem">AF+BG theorem</a></li> <li><a href="/wiki/B%C3%A9zout%27s_theorem" title="Bézout's theorem">Bézout's theorem</a></li> <li><a href="/wiki/Bitangent" title="Bitangent">Bitangent</a></li> <li><a href="/wiki/Cayley%E2%80%93Bacharach_theorem" title="Cayley–Bacharach theorem">Cayley–Bacharach theorem</a></li> <li><a href="/wiki/Conic_section" title="Conic section">Conic section</a></li> <li><a href="/wiki/Cramer%27s_paradox" title="Cramer's paradox">Cramer's paradox</a></li> <li><a href="/wiki/Cubic_plane_curve" title="Cubic plane curve">Cubic plane curve</a></li> <li><a href="/wiki/Fermat_curve" title="Fermat curve">Fermat curve</a></li> <li><a href="/wiki/Genus%E2%80%93degree_formula" title="Genus–degree formula">Genus–degree formula</a></li> <li><a href="/wiki/Hilbert%27s_sixteenth_problem" title="Hilbert's sixteenth problem">Hilbert's sixteenth problem</a></li> <li><a href="/wiki/Nagata%27s_conjecture_on_curves" title="Nagata's conjecture on curves">Nagata's conjecture on curves</a></li> <li><a href="/wiki/Pl%C3%BCcker_formula" title="Plücker formula">Plücker formula</a></li> <li><a href="/wiki/Quartic_plane_curve" title="Quartic plane curve">Quartic plane curve</a></li> <li><a href="/wiki/Real_plane_curve" title="Real plane curve">Real plane curve</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Riemann_surface" title="Riemann surface">Riemann surfaces</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Belyi%27s_theorem" title="Belyi's theorem">Belyi's theorem</a></li> <li><a href="/wiki/Bring%27s_curve" title="Bring's curve">Bring's curve</a></li> <li><a href="/wiki/Bolza_surface" title="Bolza surface">Bolza surface</a></li> <li><a href="/wiki/Compact_Riemann_surface" class="mw-redirect" title="Compact Riemann surface">Compact Riemann surface</a></li> <li><a href="/wiki/Dessin_d%27enfant" title="Dessin d'enfant">Dessin d'enfant</a></li> <li><a href="/wiki/Differential_of_the_first_kind" title="Differential of the first kind">Differential of the first kind</a></li> <li><a href="/wiki/Klein_quartic" title="Klein quartic">Klein quartic</a></li> <li><a href="/wiki/Riemann%27s_existence_theorem" class="mw-redirect" title="Riemann's existence theorem">Riemann's existence theorem</a></li> <li><a href="/wiki/Riemann%E2%80%93Roch_theorem" title="Riemann–Roch theorem">Riemann–Roch theorem</a></li> <li><a href="/wiki/Teichm%C3%BCller_space" title="Teichmüller space">Teichmüller space</a></li> <li><a href="/wiki/Torelli_theorem" title="Torelli theorem">Torelli theorem</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Constructions</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dual_curve" title="Dual curve">Dual curve</a></li> <li><a href="/wiki/Polar_curve" title="Polar curve">Polar curve</a></li> <li><a href="/wiki/Smooth_completion" title="Smooth completion">Smooth completion</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Structure of curves</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Divisors on curves</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abel%E2%80%93Jacobi_map" title="Abel–Jacobi map">Abel–Jacobi map</a></li> <li><a href="/wiki/Brill%E2%80%93Noether_theory" title="Brill–Noether theory">Brill–Noether theory</a></li> <li><a href="/wiki/Clifford%27s_theorem_on_special_divisors" title="Clifford's theorem on special divisors">Clifford's theorem on special divisors</a></li> <li><a href="/wiki/Gonality_of_an_algebraic_curve" title="Gonality of an algebraic curve">Gonality of an algebraic curve</a></li> <li><a href="/wiki/Jacobian_variety" title="Jacobian variety">Jacobian variety</a></li> <li><a href="/wiki/Riemann%E2%80%93Roch_theorem" title="Riemann–Roch theorem">Riemann–Roch theorem</a></li> <li><a href="/wiki/Weierstrass_point" title="Weierstrass point">Weierstrass point</a></li> <li><a href="/wiki/Weil_reciprocity_law" title="Weil reciprocity law">Weil reciprocity law</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Moduli</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/ELSV_formula" title="ELSV formula">ELSV formula</a></li> <li><a href="/wiki/Gromov%E2%80%93Witten_invariant" title="Gromov–Witten invariant">Gromov–Witten invariant</a></li> <li><a href="/wiki/Hodge_bundle" title="Hodge bundle">Hodge bundle</a></li> <li><a href="/wiki/Moduli_of_algebraic_curves" title="Moduli of algebraic curves">Moduli of algebraic curves</a></li> <li><a href="/wiki/Stable_curve" title="Stable curve">Stable curve</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Morphisms</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hasse%E2%80%93Witt_matrix" title="Hasse–Witt matrix">Hasse–Witt matrix</a></li> <li><a href="/wiki/Riemann%E2%80%93Hurwitz_formula" title="Riemann–Hurwitz formula">Riemann–Hurwitz formula</a></li> <li><a href="/wiki/Prym_variety" title="Prym variety">Prym variety</a></li> <li><a href="/wiki/Weber%27s_theorem_(Algebraic_curves)" title="Weber's theorem (Algebraic curves)">Weber's theorem (Algebraic curves)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Singular_point_of_a_curve" title="Singular point of a curve">Singularities</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Ak_singularity" title="Ak singularity">Ak singularity</a></li> <li><a href="/wiki/Acnode" title="Acnode">Acnode</a></li> <li><a href="/wiki/Crunode" title="Crunode">Crunode</a></li> <li><a href="/wiki/Cusp_(singularity)" title="Cusp (singularity)">Cusp</a></li> <li><a href="/wiki/Delta_invariant" class="mw-redirect" title="Delta invariant">Delta invariant</a></li> <li><a href="/wiki/Tacnode" title="Tacnode">Tacnode</a></li></ul> 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Elliptic curve - Wikipedia