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History of Special $L$-value of elliptic curves over $\mathbb{Q}$ of rank $3$

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2026-08-13 22:07 bmatschke how well the digits are known: assumed-bound (interval widened by a hand-chosen 4 ulps (blur_real_interval)) current
2026-08-09 09:14 flattening entries rewritten as records with named parameters
2026-08-09 08:35 data-repository import the current state of the data repository reviewed
2021-05-08 14:22 bmatschke from the data repository, e76a6b04

The first version, 2021-05-08 14:22

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+Title: Special $L$-value of elliptic curves over $\mathbb{Q}$ of rank $3$+Definition: The special $L$-value of an elliptic curve over $\mathbb{Q}$  of (algebraic)+  rank $r$ is  the $r$'th derivative of  its $L$-function $L(E,s)$ at $s=1$. Here+  the arithmetic normalization of $L(E,s)$ is used CITE{LMFDB-special-L-value}. This+  table lists $L^{(3)}(E,1)$ for elliptic curves over $\mathbb{Q}$ of rank $3$ for+  the first few conductors.+Parameters:+  N:+    title: conductor of $E$+    type: Z+  c4:+    display: $c_4$+    title: invariant $c_4$ of a minimal model of $E$+    type: Z+  c6:+    display: $c_6$+    title: invariant $c_6$ of a minimal model of $E$+    type: Z+Comments:+  comment-optimal-curves-only: As $L(E,s)$ only depends on the $\mathbb{Q}$-isogeny+    class of $E$, the table lists at most one representative of each isogeny class.+  comment-BSD: The Birch and Swinnerton-Dyer conjecture CITE{WikiBSD} states that+    $\frac{L^{(r)}(E,1)}{r!} = \frac{|\text{Sha}(E)|\Omega_E R_E \prod_{p|N} c_p}{|E_{\text{Tor}}|^2}$.+  comment-redundancy-of-N: The conductor $N$ can be deduced from $c_4$ and $c_6$.+    We display it for convenience.+  comment-E-from-c4c6: 'A model of $E/\mathbb{Q}$ is given by $E: y^2 = x^3 - 27 c_4+    x - 54 c_6$.'+Formulas: {}+Programs:+  program-sage:+    language: Sage+    code: "numbers = [E.lseries().taylor_series(series_prec=4)[3] * factorial(3) \n\+      \                         for E in cremona_curves([1..100000])\n           \+      \              if E.rank() == 3]"+References: {}+Links:+  LMFDB:+    title: 'LMFDB: Elliptic curves over Q'+    url: https://www.lmfdb.org/EllipticCurve/Q/+  LMFDB-special-L-value:+    title: 'LMFDB: Special value of an elliptic curve L-function'+    url: https://www.lmfdb.org/knowledge/show/ec.q.special_value?special_value+  WikiBSD:+    title: 'Wikipedia: Birch and Swinnerton-Dyer conjecture'+    url: https://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture+Similar tables: ''+Keywords: ''+Tags:+- number theory+- elliptic curves+- special values+- L-function+Data properties:+  type: R+  complete: 'no'+  sources:+  - CITE{LMFDB}+Display properties:+  number-header: $L^{(3)}(s,1)$+  group parameters:+  - - N+  - - c4+    - c6+Numbers:+  '5077':+    336, -5400:+      number: '10.39109940071580413875185051036091706972635637565700927972447903053935197449771279993813633135361004'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5077a1}[Curve in LMFDB]+  '11197':+    273, 999:+      number: '17.22951452514339009969603683718323424790391601327111423152123992617796285275908674932923908677478565'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/11197a1}[Curve in LMFDB]+  '11642':+    777, -20709:+      number: '18.99557922283332410650894683705725710744754108853102831314035440043687047554565861540536754265728821'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/11642a1}[Curve in LMFDB]+  '12279':+    496, -7640:+      number: '17.51419486047049899536915522059318299180016332278159868544826648238636340691028561550121659377053064'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/12279a1}[Curve in LMFDB]+  '13766':+    1081, -38125:+      number: '21.72163474567358885617993090781346296310418424123657468731850875260833722328318083996378922817539381'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/13766a1}[Curve in LMFDB]+  '16811':+    48, -5400:+      number: '16.23197048041948089465723971837326817960114540311759130906259420148634099636506658368386062697799641'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/16811a1}[Curve in LMFDB]+  '18097':+    481, -8945:+      number: '24.64166314374359266849158660872113725866941324353237862040697733572611663253780725562359838393747284'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/18097b1}[Curve in LMFDB]+  '18562':+    937, -27541:+      number: '24.58306496204720479947405648951902687944870264691516992162468679478239187054272824588181009389405609'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/18562c1}[Curve in LMFDB]+  '18745':+    7024, -591832:+      number: '19.47785333642883719200737373512510751161036183310773239585071303541969763804261698578742787864831072'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/18745a1}[Curve in LMFDB]+  '20888':+    2496, -86400:+      number: '29.75389763858687895103611410704151473788012579323589923846047789245294672907571180999242449952184455'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/20888a1}[Curve in LMFDB]+  '21443':+    241, -7145:+      number: '25.66040174439519489047747662567993598902756359200063364758235100681275717167154985315065649156277313'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/21443a1}[Curve in LMFDB]+  '21858':+    1561, -63485:+      number: '28.30862501021523389953282334649424456854374757456757757671202501374306316046360675377808022909636012'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/21858a1}[Curve in LMFDB]+  '22481':+    -303, -3321:+      number: '29.80285702756469961290369111908185666802466051625796769524880378423694961959475200078094139819829070'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/22481a1}[Curve in LMFDB]+  '22696':+    5056, -357760:+      number: '33.23846014597736647332669728535625511929220006348027173771851386745727141416866265046828128283884579'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/22696a1}[Curve in LMFDB]+  '24301':+    2113, -97345:+      number: '24.79497677734949220883255753995984536567443655753787181095413367004903612971076285663895066622259499'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/24301a1}[Curve in LMFDB]+  '24546':+    1897, -84149:+      number: '30.38980280932429985469186721686963082689930437676641013911891198628950861789676418150726113143888518'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/24546a1}[Curve in LMFDB]+  '24646':+    601, -11485:+      number: '26.69441954137114737270384907863342134421276391807531806547980729502065880022257874566041897096156414'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/24646b1}[Curve in LMFDB]+  '25071':+    1168, 20584:+      number: '23.59894904496478392251010018707209105068510471607754991090167374321698981269571020481524345807947026'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/25071a1}[Curve in LMFDB]+  '25383':+    1552, -60056:+      number: '22.71317605383302707604079176628579646586389463979956494523274233177085651433924638272617868387614750'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/25383a1}[Curve in LMFDB]+  '25451':+    2593, -137105:+      number: '28.63537661781538558571208989207779840332589781782211074994772937226708502105225192331811549895523020'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/25451a1}[Curve in LMFDB]+  '25751':+    1873, -84025:+      number: '25.90284029216747027749915291492761942384020813497072899053843374790440845532890978916580107566634243'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/25751a1}[Curve in LMFDB]+  '26171':+    2352, -114264:+      number: '17.34483849922828192308555319609904415933071164526352499668863868813384072751831942669918420055657770'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/26171a1}[Curve in LMFDB]+  '26198':+    777, -10341:+      number: '23.82796023291876584953017053582729500192361314211256632692384709902196974977907880867685272267612696'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/26198a1}[Curve in LMFDB]+  '26284':+    448, -16480:+      number: '36.41165983303400173705918557265884880720491021833541277501219625008770238061646982289950195253825187'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/26284a1}[Curve in LMFDB]+  '26743':+    -287, 4751:+      number: '25.29467302905279175342741267692458134728491639435043327988333360959591230109166096615875311884120528'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/26743a1}[Curve in LMFDB]+  '27262':+    11361, -1128465:+      number: '42.61727627748339966732486403769423842358732977022344814716553718529201030704442867363606396109649653'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/27262c1}[Curve in LMFDB]+  '27382':+    457, -901:+      number: '28.35553580753106450276075509993963835661481946025606042223052041338345216499082553404835211002705121'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/27382a1}[Curve in LMFDB]+  '27448':+    3792, -236736:+      number: '32.41846940383448287420958358276921832193943837531743537247225572209367632280020259348914136831731382'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/27448d1}[Curve in LMFDB]+  '27584':+    192, -55296:+      number: '32.37398514225308436152013055131446114203066191387726033872505593530748357351815858375988350018235631'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/27584bd1}[Curve in LMFDB]+  '27746':+    553, -23509:+      number: '28.96885277219437460761015648209693040405202761824599152284113216502629599315507483701797601275107354'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/27746a1}[Curve in LMFDB]+  '27747':+    15696, -1975320:+      number: '25.81302691028774621553297740896109381243935096382725175480935423458088549009511553544096501084530505'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/27747c1}[Curve in LMFDB]+  '27773':+    993, -32049:+      number: '22.54047481216334771338510818813260913422574140461250469716199815455342974774099963224723099802177167'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/27773a1}[Curve in LMFDB]+  '27808':+    17472, -2294784:+      number: '32.21627996413513934597582897036390794004359770383642703900222265582991835335357819699404670935130230'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/27808a1}[Curve in LMFDB]+  '28042':+    921, -38205:+      number: '24.89330724858652980132001323651036066257794879621731291991795850412819400584440811133614079591002900'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/28042b1}[Curve in LMFDB]+  '28498':+    633, -18765:+      number: '26.83297717411920823540961637131892399567868245134452858716751270436702501793075225665195918467348658'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/28498a1}[Curve in LMFDB]+  '28571':+    208, -7640:+      number: '24.06873614466021532133630798308975645358923652221871874355310122684886912332904633036750665957788943'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/28571a1}[Curve in LMFDB]+  '29157':+    11536, -1097560:+      number: '29.02584690489948566990539773561221315302057791505259987312350602580694825198341521329010797769213005'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/29157b1}[Curve in LMFDB]+  '30064':+    7312, -627904:+      number: '37.64393602375145991259255323932895192855039054682337693121594738309857528419420866179702941297269473'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/30064c1}[Curve in LMFDB]+  '30148':+    1776, -73440:+      number: '32.25595696473959956656803043375130803324605942010995458239625736323732789140715808731997572486560020'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/30148b1}[Curve in LMFDB]+  '30376':+    1216, 10880:+      number: '37.11982689877911149763731143657096051615685506987912202641096328951379292101249881510848945519245340'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/30376a1}[Curve in LMFDB]+  '30446':+    1897, 36811:+      number: '32.20339802002112426165748092806431321589945130025663634574982636919093883323403849160074484126803346'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/30446a1}[Curve in LMFDB]+  '30487':+    1344, 12744:+      number: '31.78788073671751223287276005089179172194491308937646688249220410274903273748412781564623239569065014'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/30487a1}[Curve in LMFDB]+  '30767':+    1488, -52056:+      number: '20.02056760671028111486750476410041512480088220386334662677388622237647001726681315388721857116536493'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/30767d1}[Curve in LMFDB]+  '30815':+    784, -27352:+      number: '22.39238867055076195355150727319616327445801838485049645980299416479052465422001474834722563801053272'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/30815a1}[Curve in LMFDB]+  '31478':+    217, -10909:+      number: '29.39379272123397797703537884363855784352654529118338360810106054666056138868155958432449779082028341'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/31478b1}[Curve in LMFDB]+  '31737':+    1648, -68120:+      number: '25.29775935254938068256457908145351010935413934755910658039874930925863835384615895013627242932476603'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/31737a1}[Curve in LMFDB]+  '31814':+    -119, -10405:+      number: '27.99164137240807611193798471764149572156425952536606927007618789897181575590794636785160359421505570'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/31814a1}[Curve in LMFDB]+  '32192':+    2496, -138240:+      number: '33.13696925057679976130473555055284320023867981221797302516942847170564295695913020833966357510976975'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/32192e1}[Curve in LMFDB]+  '32244':+    1456, -61280:+      number: '41.46800381861746039885927209477237169978360735128055716711120234099575355104323117719179978002651198'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/32244c1}[Curve in LMFDB]+  '32276':+    832, -18784:+      number: '37.99761431812338976355444770928926037353210551941661412311577026950059561450705154727976586775486644'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/32276b1}[Curve in LMFDB]+  '32336':+    1552, -12736:+      number: '38.39000497579469295236318309852285617290067105340484660839623577169612542028000959528347270535017710'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/32336b1}[Curve in LMFDB]+  '33424':+    14992, -1831616:+      number: '43.21845972156542025518503717759144107266933192238380421484509972655851275118343121168312577836099446'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/33424o1}[Curve in LMFDB]+  '33428':+    2176, -81280:+      number: '37.31433246726619136865129857395856120088462360255622626436812170426853492764570556925021268745174994'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/33428a1}[Curve in LMFDB]+  '33509':+    1072, -34264:+      number: '22.17888537511989073487622651467120552971670115271161642537721171384000285447665910037433229205916061'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/33509a1}[Curve in LMFDB]+  '33609':+    11137, -1174049:+      number: '29.70556821348130493263329937153733755857486900300432358072949798649975532390011157994405723070137909'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/33609a1}[Curve in LMFDB]+  '33709':+    2289, 46359:+      number: '24.78025949988191823488251607968848384790858064895381011688910576802234338302683852402819486395416208'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/33709a1}[Curve in LMFDB]+  '33914':+    1321, -245845:+      number: '29.18383145934485863033987674984491427808742920558549953230685023794630616484274971156684175271489996'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/33914a1}[Curve in LMFDB]+  '34415':+    -431, -85753:+      number: '28.45677059471286278182607192157561106009605405363164959078100043342362561103427087694506219693449694'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/34415a1}[Curve in LMFDB]+  '34862':+    2721, -286065:+      number: '46.78278408883760422860270561092373289588614288549327524637373955355265554013656675268941277595973672'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/34862a1}[Curve in LMFDB]+  '35003':+    -848, 20456:+      number: '22.08833304916740878081799976617103196815753861232762238715246723158169347273907020868671343119007919'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/35003a1}[Curve in LMFDB]+  '35083':+    208, -8344:+      number: '23.77941735887846531633461642538466711931192101805844808724138467752225622210671855737221192700322480'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/35083c1}[Curve in LMFDB]+  '35234':+    8713, -712421:+      number: '35.38103867927958168752769122375286320811625290094155611229732887060176948908599137121190920518277288'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/35234a1}[Curve in LMFDB]+  '35243':+    1057, -19729:+      number: '26.06210877862169241557764519634383565580312808856162096650984796310265954231246561780072160414708227'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/35243b1}[Curve in LMFDB]+  '35882':+    3081, -196965:+      number: '27.02512781490412095338776298199878360920469003574754448391038046182589252413047458028744122403021182'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/35882b1}[Curve in LMFDB]+  '36094':+    7761, -989145:+      number: 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