History of Special $L$-value of elliptic curves over $\mathbb{Q}$ of rank $2$

who	what
2026-08-13 22:06	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 13:57	bmatschke	from the data repository, 6667cce7
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+Title: Special $L$-value of elliptic curves over $\mathbb{Q}$ of rank $2$+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 $L''(E,1)$ for elliptic curves over $\mathbb{Q}$ of rank $2$ 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=3)[2] * factorial(2) \n\+      \                         for E in cremona_curves([1..1000])\n             \+      \            if E.rank() == 2]"+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: $L^{(2)}(s,1)$+  group parameters:+  - - N+  - - c4+    - c6+Numbers:+  '389':+    112, -856:+      number: '1.518633000576853540460385215789444038156195032989848703171610185095982145495561104880431870004657718'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/389a1}[Curve in LMFDB]+  '433':+    1, -865:+      number: '1.894041561731629066979452801961502506408696716292726162536379245600233700458618579384375246287372150'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/433a1}[Curve in LMFDB]+  '446':+    201, -2565:+      number: '1.880565209379241373883636368301211885704022011375167129954306221603446752406944334869494596357306112'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/446d1}[Curve in LMFDB]+  '563':+    721, -19385:+      number: '2.269111834006667743465417706491576141965345126842304484135594884125798300031340420753154889534591295'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/563a1}[Curve in LMFDB]+  '571':+    208, -3160:+      number: '1.806321033943328939862297250514478757278446062286306066446067993632970853294285861276927597601484497'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/571b1}[Curve in LMFDB]+  '643':+    193, -2881:+      number: '2.296523473097996561680680051327619762430111774019924214655891731199196701850998342747579464809305466'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/643a1}[Curve in LMFDB]+  '655':+    624, -15768:+      number: '1.776384221654996543349013518357039198021468826728728775772925567710510491949287954050967128643730936'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/655a1}[Curve in LMFDB]+  '664':+    336, -8640:+      number: '2.748405352270020422946565221908210281341466718742019045062019222985289325253394422092405687294576033'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/664a1}[Curve in LMFDB]+  '681':+    16, -1880:+      number: '2.052542294788023656746967696595152454266296165974990116569500644039495128825578505628180437351514050'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/681c1}[Curve in LMFDB]+  '707':+    592, -14104:+      number: '1.947702692054184944180993877762422839765419273640849746191394752647876270063285163695837905692584706'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/707a1}[Curve in LMFDB]+  '709':+    112, 424:+      number: '2.107403307618986744343066621066513437403620476249984341046684994580141571945389853580757044634384129'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/709a1}[Curve in LMFDB]+  '718':+    217, -541:+      number: '2.503562305537786026658086192492903036161690581267179665365799765642431623784585341018531090614443899'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/718b1}[Curve in LMFDB]+  '794':+    121, -2125:+      number: '2.589083824262024385267363379402853853599605902655720345539870561668492383693386883777269208711458670'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/794a1}[Curve in LMFDB]+  '817':+    -32, -5176:+      number: '3.132940778845538517148479708838798693142024961139998564806098240422510727444075489023559845132395874'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/817a1}[Curve in LMFDB]+  '916':+    192, -864:+      number: '3.195096301121879513306308519366195663309952550740069516355403188184368141100935250088512871726534937'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/916c1}[Curve in LMFDB]+  '944':+    912, -29376:+      number: '3.243235165770318152472666015010585981392393695784501989431585396673481936312505533169334691930607319'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/944e1}[Curve in LMFDB]+  '997':+    256, 3880:+      number: '3.601147456650398660956092992293048399425806443520964962898192304801853131627580559759162971800023697'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/997b1}[Curve in LMFDB]+    1168, -39896:+      number: '2.440570144195680788684452819942002783273638309108476141573768262555253562535680845165967484680779744'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/997c1}[Curve in LMFDB]+  '1001':+    9552, -943704:+      number: '1.972052899099280170044156043434055922935536668059954928981974500672549570995381552628255828398281009'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1001c1}[Curve in LMFDB]+  '1028':+    496, -10720:+      number: '3.572015034013850948261168443830346574525907401501891410720473096130944281414580006433048662191288235'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1028a1}[Curve in LMFDB]+  '1034':+    553, -13141:+      number: '2.944334147089450688613498684214578269895950986196339956700303809848248438804550718414553313540284833'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1034a1}[Curve in LMFDB]+  '1058':+    5497, -407629:+      number: '2.952040418916963589267680821021635340112347203999322090613734120752175433411669529174956420843756660'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1058c2}[Curve in LMFDB]+  '1070':+    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'3.020135756756256473459103013085972888812377370307296224448835400342845381217622164797173902545407617'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1094a1}[Curve in LMFDB]+  '1102':+    1417, -63269:+      number: '3.247697551910489797153201399829126974990125520205857275870507934074470940029516971010553451857626341'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1102a1}[Curve in LMFDB]+  '1126':+    -87, -3861:+      number: '2.794583584503614782354001520939921439404108832783789313477261772387957550345536700988283321998517558'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1126a1}[Curve in LMFDB]+  '1132':+    256, -4960:+      number: '3.728955325046394846019278657110229120495692360914191938735467011662516961650547850590626908095426067'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1132a1}[Curve in LMFDB]+  '1137':+    97, -2609:+      number: '3.093112786930884713390000066331913734555365567030651905046693618308171571172692347156730115044221855'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1137a1}[Curve in LMFDB]+  '1141':+    1297, -83161:+      number: '2.847982705591171942119770577648958651378355141985976699086867887793151658603302870632032174652176358'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1141a1}[Curve in LMFDB]+  '1143':+    1872, -77976:+      number: '2.776527068590015340041659797888198469776945656584002972777883422978610500002325973074332008039986572'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1143c1}[Curve in LMFDB]+  '1147':+    448, -5336:+      number: '3.889208559381645639128189786635347797436665466506304920747098154270134758000447925645953439121750914'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1147a1}[Curve in LMFDB]+  '1171':+    129, 351:+      number: '2.680434816135108974532183127033981056579136555640342676512727506708269979273003645039281803962298330'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1171a1}[Curve in LMFDB]+  '1246':+    57, -10989:+      number: '2.861553678177483391748447515178935849350721936821795829290715601225535507961679956503686917068034717'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1246c1}[Curve in LMFDB]+  '1309':+    1072, -38744:+      number: '2.672363227443661558550081897521614886799234952753348685380300074278731360774128835853064051055287423'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1309b1}[Curve in LMFDB]+  '1324':+    -128, -2656:+      number: '4.018557251803585701507720721835183171243112553514228409216572208731390566501330321146535891656502602'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1324a1}[Curve in LMFDB]+  '1325':+    400, 2600:+      number: '2.561546809234704993295983274559264707971481706784263887631655793574488722867557382272151747741586835'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1325e1}[Curve in LMFDB]+  '1431':+    1377, 28431:+      number: '3.665942762631463591599072621044909774678589523551529982376993616912450360461809346886551906264534486'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1431a1}[Curve in LMFDB]+  '1436':+    592, -6976:+      number: '4.164705662775022478239382819302256690354775101593860004857239753093523595717683034855125166857322752'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1436a1}[Curve in LMFDB]+  '1443':+    433, -8585:+      number: '3.406961745279545580802375030849363625744469638447516376304939476550058914637941095552185951020115563'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1443c1}[Curve in LMFDB]+  '1446':+    217, -5021:+      number: '3.601183472869611257115091504775505421269752879132575314670631870472785540288048615780426720748288899'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1446a1}[Curve in LMFDB]+  '1466':+    2001, -81945:+      number: '4.912207291645502188834497160988606394044377765394319936819861221947496328389915472705869251945309431'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1466b1}[Curve in LMFDB]+  '1477':+    289, -6481:+      number: '3.210157024258067718808512448954577156165204729989147172774574908866929872319126409462288379538175932'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1477a1}[Curve in LMFDB]+  '1480':+    1344, -44928:+      number: '3.960512887404247950557587747449704223349863406684839610062936972052137036154176807952537114126727848'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1480a1}[Curve in LMFDB]+  '1483':+    -80, -1432:+      number: '2.657964170211268317791709754318858565613605850475420339540920214897225174571183245977747167425084657'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1483a1}[Curve in LMFDB]+  '1525':+    1585, 56375:+      number: '3.268209134955411740901593764287160970156084174848252041875268154070242676617294744795768381921148367'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1525c2}[Curve in LMFDB]+  '1531':+    672, -17496:+      number: '4.691007143066202937894706028930593100717817226948166601659469219434864032525436280181818945889605626'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1531a1}[Curve in LMFDB]+  '1534':+    -231, -33021:+      number: '3.135043256242203191133221762716170901289626432657339544957276592368661280827911423223958022220213177'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1534b1}[Curve in LMFDB]+  '1570':+    169, -5653:+      number: '3.543163368541703026367004551300196494387068095522276902360693625014923097664148468240211635167345829'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1570b1}[Curve in LMFDB]+  '1576':+    448, 1664:+      number: '4.357224810831051154599040417028856551870269147083277438786671452303035932290201417150629524037307204'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1576a1}[Curve in LMFDB]+  '1591':+    3408, -477144:+      number: '3.186117540011986116936161414040403433736295946037044779691009914310527630837305860963922201106116223'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1591a1}[Curve in LMFDB]+  '1594':+    1281, -59265:+      number: '5.079913340658189329287647841839340917496457737918468035540276629066062856602418490928258074919419674'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1594a1}[Curve in LMFDB]+  '1608':+    1216, -45440:+      number: '4.660722571762818129998450281692306965955828999078381856838008098966076657522364324968143809349756589'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1608a1}[Curve in LMFDB]+  '1611':+    432, -17496:+      number: '3.141205712540697813493709732299689851127124922115374892232349773185058110951889424820129660639497368'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1611d1}[Curve in LMFDB]+  '1613':+    160, -1144:+      number: '4.231196348140473769474084444078949930044578742380256262167303339442098975040012476982606150286302270'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1613a1}[Curve in LMFDB]+  '1615':+    10321, -1045369:+      number: '3.408641492481323945275179912945018938098448531283440119559369959634743446763800913609581697501621752'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1615a1}[Curve in LMFDB]+  '1621':+    177, -2889:+      number: '3.862255766693157794820652767426909983481530048179104201894902834850593129779010388907945409175522136'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1621a1}[Curve in LMFDB]+  '1627':+    145, 487:+      number: '3.694632516359735257695281349584604831060989170387038367941803761985164447310837533542003008976798604'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1627a1}[Curve in LMFDB]+  '1639':+    273, -4185:+      number: '2.891274188929709583353539197256436885545459808339563175716284329279514120293312098856645144155267198'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1639b1}[Curve in LMFDB]+  '1641':+    208, -4184:+      number: '2.954030456334073900609596450921222804909858300719307567114728598919115410106320780453984266638434327'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1641b1}[Curve in LMFDB]+  '1642':+    73, -4805:+      number: 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comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/3672i1}[Curve in LMFDB]+  '3681':+    -1152, 4104:+      number: '6.731127928627093919966410083917105855783558545965463858246386398693860361520177469027149593020187889'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/3681a1}[Curve in LMFDB]+    16560, -2134296:+      number: '4.511502230707749869766234805648896746424060303235010649672738287789026599506441760982887555922151522'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/3681b1}[Curve in LMFDB]+  '3684':+    448, -12896:+      number: '6.549461559129491521779426611070792083576371545326972020414420050428953603447958007673297732296116054'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/3684a1}[Curve in LMFDB]+  '3686':+    7153, -758089:+      number: '7.930736400797579334660579953599887432528484331932642972047428874883089365619011407244797957834746043'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/3686e1}[Curve in LMFDB]+  '3687':+    577, 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  comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/3776s1}[Curve in LMFDB]+  '3779':+    -96, 2376:+      number: '4.758864417432643014952517992122565050018653175604053902061423851720905555075684197273577105030116809'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/3779a1}[Curve in LMFDB]+  '3782':+    6297, -501309:+      number: '5.611594588576598336251699390542016355879406514401230846827871052099734757207503976226713498903980792'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/3782a1}[Curve in LMFDB]+  '3784':+    -9536, 470656:+      number: '6.820493056769359514559087264405882923364883806899572436554907116536634322307014199655004972917456611'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/3784f1}[Curve in LMFDB]+  '3786':+    -167, -3869:+      number: '5.260186916149826156954082289879099996420835106226533040621433451123882847380252579073471260581060961'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/3786a1}[Curve in LMFDB]+    9841, 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  comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/4136b1}[Curve in LMFDB]+  '4139':+    -176, -1304:+      number: '4.310891145991977496399687960476970772536784278467127729295066739472558732108063197394344530170416032'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/4139a1}[Curve in LMFDB]+  '4144':+    70288, -18216640:+      number: '6.375833865926902409605427600057095446420654982129575586983595912606076971046556863077876826037357895'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/4144e2}[Curve in LMFDB]+  '4178':+    457, 6139:+      number: '5.270515123111351770815882244041092395350354612536891611296175986997459591458840976857889010802768658'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/4178d1}[Curve in LMFDB]+    2577, -48249:+      number: '7.397114777816488171559781948671729972652815490445272637690761000165559472108472763723949096099600994'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/4178e1}[Curve in LMFDB]+  '4182':+    5761, 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comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/4547a1}[Curve in LMFDB]+  '4554':+    2313, -20709:+      number: '6.152604112591958164657855339219465465254586385814454966226104553806674219810366167048227234611311044'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/4554b2}[Curve in LMFDB]+  '4564':+    -48, 22464:+      number: '7.836742542161434199616395434988331872722789756481540083795645344684133231430277202904717573601858468'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/4564a1}[Curve in LMFDB]+  '4569':+    1393, -28009:+      number: '6.207278059676952564028310402381912851981865836714262207881016139716683511990797141095022680356726701'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/4569b1}[Curve in LMFDB]+  '4572':+    1152, -3456:+      number: '7.838282265323715315950251563793703060758588041745272809129891633781951556471563845327055702002968178'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/4572b1}[Curve in LMFDB]+  '4578':+    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comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/4623c1}[Curve in LMFDB]+  '4629':+    208, 1000:+      number: '4.274727423392353608536197736411687839003224187355696802759073866597295854056864384866254227198397911'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/4629a1}[Curve in LMFDB]+  '4630':+    -231, -1917:+      number: '4.549640872753996967322704595658359915154940342871347023926442487300635461076928450048541989550052944'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/4630b1}[Curve in LMFDB]+    7561, -662309:+      number: '6.057845576826380255988960453509717904987680464664775355427212288474641771952158723173194779417141947'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/4630c1}[Curve in LMFDB]+  '4632':+    30736, 5220800:+      number: '8.408954299543414390357557330923373529302667147362294994059314801469250369194869331539966421993254372'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/4632f2}[Curve in LMFDB]+  '4652':+    768, 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comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6369d1}[Curve in LMFDB]+    -28943, -7374889:+      number: '6.803900642760099383559783625587219788776048533659502470895428138015170956161336703856311208131474840'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6369g1}[Curve in LMFDB]+  '6370':+    17689, -3030013:+      number: '5.744079778753691398693071183739905194693424620873558295015607169419382627221194950344260621057360449'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6370a1}[Curve in LMFDB]+  '6372':+    1008, -25056:+      number: '8.864741234049721092195243384353358447311928377451973859760750567553123108538260662800356822308223103'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6372c1}[Curve in LMFDB]+  '6376':+    6160, -484928:+      number: '8.361848960920797417358411538952843160025769154105272174363470153659844700484066742354393760382073221'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6376a1}[Curve in LMFDB]+  '6378':+    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comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6683a1}[Curve in LMFDB]+  '6688':+    5008, -927296:+      number: '8.160942447600291273597536996382091605056636682515880424824004447199016199970964400996099199783065882'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6688b1}[Curve in LMFDB]+  '6691':+    -47, -3385:+      number: '4.429444980021823955131094954727512739612357955956119731498880579242236397123393100175692427267995099'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6691b1}[Curve in LMFDB]+  '6696':+    -15552, -2146176:+      number: '9.132409528154604915114565208716983815214161899894325291955127130927361703970066184291472283030002014'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6696d1}[Curve in LMFDB]+    146448, -56033856:+      number: '9.092622845610448845779865152700223349108529393701784424677630975082772082203388657545196869510617719'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6696e1}[Curve in LMFDB]+  '6700':+    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'10.49916722044226304325843198238209224531996074167226895705153814305429486334203008073600222113301770'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6751b1}[Curve in LMFDB]+    40816, 10636840:+      number: '6.087538838155109101053950372684496603189313887251148419789259639700792782687208775893822639274204866'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6751c1}[Curve in LMFDB]+  '6760':+    2704, -5408:+      number: '8.261405387310495785095457508507846603258217355577153476967519885809571668502372453707817865651381592'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6760h1}[Curve in LMFDB]+  '6766':+    1641, 66123:+      number: '5.526526229779261306137801208084435559217323143162892701670568713614736899279487304221266944140307501'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6766b1}[Curve in LMFDB]+    3697, -196361:+      number: '10.69492043358622771279238872035463817736580717754316007947522493065953757978601960306709790453022903'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6766d1}[Curve in LMFDB]+  '6768':+    1152, 31104:+      number: '8.865314742767454834030509659532318105185995205173969495519409539901256115754672656855343858515816508'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6768i1}[Curve in LMFDB]+  '6770':+    1209, -47277:+      number: '5.085847402931165233937888650238475166815438244940343721155018215564377628348598320460335680712988429'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6770a1}[Curve in LMFDB]+  '6771':+    1825, 56879:+      number: '7.609550456660935290951287677093661165243127329225355321754213673565217346473915899338733659183753379'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6771c1}[Curve in LMFDB]+  '6776':+    86592, -25508736:+      number: '6.311545962000490446284601173782980716577221618834967997570732304713378282601069534694441513896461320'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6776b1}[Curve in LMFDB]+  '6782':+    25297, -4023449:+      number: '10.59135421560168048712216239761205160743163832229887397865102759348981165525701805843884766328615254'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6782b1}[Curve in LMFDB]+    4737, 101439:+      number: '8.561741160128613359791436920326998404982039906101972203381049680938226135420439763641658159358347060'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6782c1}[Curve in LMFDB]+  '6784':+    5248, 2087936:+      number: '10.04805710146596476502804750997319424536257637763167694446983951175687187653855275578531331979575856'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6784e1}[Curve in LMFDB]+    2752, -146944:+      number: '7.092922323746365595311929290720540625770461385406675030379379380611570192899848058240871222094841658'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6784g1}[Curve in LMFDB]+    640, -41984:+      number: '8.598040098091475042112151081942747311591532315801605733255006772521463437411187005602505761207968045'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6784i1}[Curve in LMFDB]+  '6789':+    141424, -54971416:+      number: '5.981794978096569143203811217862172555501286809591017239569994155737709061676374238042783234675275719'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6789b1}[Curve in LMFDB]+  '6792':+    1120, -45152:+      number: '8.504514157923860407701742917740834726923580181510122397622374510502344401543448404273217317850362383'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6792a1}[Curve in LMFDB]+  '6794':+    6937, 576899:+      number: '7.749959424705479357022263412670842297535782239335348123421709782330991680098709696556983214145181479'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6794a1}[Curve in LMFDB]+    22377, -3344949:+      number: '4.486483224789088878315407918482249384680536220288294375054314413102630996092098148118697215429494111'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6794b1}[Curve in LMFDB]+    158257, -62924201:+      number: '10.73391135153230029228025946651158786120049372047073271809116369432643296866206181560113126672682979'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6794c1}[Curve in LMFDB]+  '6807':+    1504, 58024:+      number: '7.553829672622566387975248790530928272360386094710533704407162229226145115566809015272916007905466206'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6807b1}[Curve in LMFDB]+    241, 1495:+      number: '4.714120286015842552030734568267520505368538908465351725405323164429062439523941514416956071909075019'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6807c1}[Curve in LMFDB]+  '6808':+    25408, -2132864:+      number: '8.145258872248201903072595855012602514903743264150192626605981081247965094492631498461193528640803678'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6808c1}[Curve in LMFDB]+  '6811':+    448, -2744:+      number: '7.218412894579181015589084343585815466656405593315306089235023123457189422612869477193865644251994818'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6811d1}[Curve in LMFDB]+    385, -11809:+      number: '5.380684410316745594061957173041280814797282641657761729049905980144694070602729770072148415396098857'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6811g1}[Curve in LMFDB]+  '6813':+    7488, -641304:+      number: '8.298011306732331375023428638658061143931814109545506693578721651813842413036516731775887393854925000'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6813a1}[Curve in LMFDB]+    2736, -109080:+      number: '4.850203045933278167992095686141809196033437719533738887008491943479200153636277694411247450341535624'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6813c1}[Curve in LMFDB]+  '6814':+    -455, 19:+      number: '6.422344781567448154992778749369607161869269611158988213076528603958794997842394599444510310085370660'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6814b1}[Curve in LMFDB]+  '6822':+    19017, -2623941:+      number: '6.875869739429735747742235075779553435546657220166551761836328872792748258247779504514842923386348813'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6822b1}[Curve in LMFDB]+  '6835':+    529, -9433:+      number: '4.622370176921593805003966031460770135741214350513730405008909841411989565660884464455399174987861906'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6835a1}[Curve in LMFDB]+  '6838':+    20329, -2897461:+      number: '5.783931908543713638205664339161998445672908373220311793434199368618216119949462917551083950657081939'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6838a1}[Curve in LMFDB]+    1977, -61101:+      number: '6.833686680438926858588400328447099421505659854543705996320571741834153990120738349025995201653861971'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6838c1}[Curve in LMFDB]+    2929, -157321:+      number: '10.10106321305573548799344619124224419388475670418434742603285759950133560291121472085783281469738546'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6838f1}[Curve in LMFDB]+  '6848':+    640, -21248:+      number: '8.603055432418657586530283193707114696346116762431265042051670516251650805197227716267013544782329013'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6848g1}[Curve in LMFDB]+    2112, -124416:+      number: '8.911383727583783491174399534911624526333458632523645459763390539984615830491307788834874664140877051'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6848m1}[Curve in LMFDB]+    30208, -5250304:+      number: '10.09227021480927149771951708300783195148536802174211600454187389199056227247885055409589467366656870'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6848p1}[Curve in LMFDB]+    -4544, 55808:+      number: '7.792568964709821485535445766415859167267708606451484059149533888923253303374792363678791662497216172'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6848s1}[Curve in LMFDB]+  '6850':+    -455, 19475:+      number: '5.969935477352662464996215727276359908848758714746418473467505775072354017574835097289740785997028243'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6850d1}[Curve in LMFDB]+  '6853':+    -4583, -118621:+      number: '12.16534475105652474635005649139481580886262149739059277863411419711921437616810131807318928445401560'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6853b1}[Curve in LMFDB]+  '6862':+    54601, -12758597:+      number: '6.918945332334955076334892230269964782532818447831294177601413977245769326160364216791911403994375939'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6862b1}[Curve in LMFDB]+    -23, 9739:+      number: '5.996656354501666662492988450457097273942931808243899202368197302969139343540155783418946658336463127'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6862c1}[Curve in LMFDB]+  '6864':+    -8048, -172736:+      number: '7.891269288492777862817184986687926252917593004489537193298275273161828862062272687464623350462316359'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6864e1}[Curve in LMFDB]+    473872, -326776256:+      number: '7.671493733400643992129735962112153123127367351179147134065588374143888852366058531098307481598695607'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6864s2}[Curve in LMFDB]+  '6867':+    1758033, -2330990649:+      number: '6.625631961503684528870205052247393626782252358665447438341565201897791875074533031386864301817043534'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6867c4}[Curve in LMFDB]+  '6868':+    1920, 79488:+      number: '6.756580572151655656791102682131367889970877485101946423730270754288453211155015508771467091363031559'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6868a1}[Curve in LMFDB]+  '6872':+    1984, -90496:+      number: '7.640865301244875498880978871610770439096773645530470456619908455716013337537102564080049328439394717'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6872a1}[Curve in LMFDB]+  '6894':+    3465, -203877:+      number: '7.507438479748281807761329266104869681576011189473284662426629353026878097555247293240544011791516043'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6894b1}[Curve in LMFDB]+    10161, -554985:+      number: '11.74444927192609431749138496882525739204408003916802090505579412429136613930218504488030475410207016'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6894m1}[Curve in LMFDB]+  '6896':+    1552, 885824:+      number: '9.964390529747911810779321015156027358715467059119272818642892228196172376513262424581432984652650047'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6896e1}[Curve in LMFDB]+    6672, -547776:+      number: '5.968460988358999560739667958693693801781448817097282727539863405160513993107031653824944473699377277'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6896f1}[Curve in LMFDB]+  '6902':+    22017, -14024961:+      number: '8.705305089843526852590089476341385926540158983906081152198352755317818256571665349619120042291988320'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6902c1}[Curve in LMFDB]+  '6903':+    95904, -11704824:+      number: '8.484406470587220562750299994598349080340421580800124648415095252548017222921561839669758769029764351'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6903a1}[Curve in LMFDB]+    432, 6696:+      number: '4.761652473259126460049280877443926853988833506289084508543881139111833642672216187817714832692559548'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6903b1}[Curve in LMFDB]+  '6905':+    -959, -81089:+      number: '6.482709509663148914338879828599435706748456980969590562197305796415065876343266102073336998147961268'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6905a1}[Curve in LMFDB]+  '6912':+    2592, -186624:+      number: '8.869937252059188640692214353223383482592549873038725222564888690884239841659948428014620383976692340'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6912l1}[Curve in LMFDB]+    -288, -13824:+      number: '9.074668223292105505146077574401342699879588589036933941152188417648307235170115253732775820419692284'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6912q1}[Curve in LMFDB]+  '6916':+    688, -25696:+      number: '7.324823767473346903239904212412468327909542446159697400314605713572148058659444486357870871667869049'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6916a1}[Curve in LMFDB]+    224368, -121429216:+      number: '7.235024507633324277773090897801969283663184974571300453257661733711986711281531762855614708674069814'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6916d1}[Curve in LMFDB]+  '6918':+    13441, 1522495:+      number: '10.61416444878646058771536462370632589993444996740153976369231388874454596954704482442578738353035309'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6918a2}[Curve in LMFDB]+  '6921':+    37728, -7344216:+      number: '8.214272389114575570045223043477933648162904941811119615196898884040377443125361731984650087367577549'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6921a1}[Curve in LMFDB]+    -720, -50328:+      number: '5.657701475296221712771918583828667459650128776501702557526036599882281765250923395865542357914392976'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6921c1}[Curve in LMFDB]+  '6945':+    -55424, 31127176:+      number: '7.959843145000582984754124775518175305921112587682174678196904177293904773637514344974392474908140483'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6945b1}[Curve in LMFDB]+    976, -42776:+      number: '5.305331092580721487506862019494144838324880408546520238716594488280566486300450701810574638017212077'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6945c1}[Curve in LMFDB]+  '6952':+    10432, -21113216:+      number: '7.871056784756292409374782378110564961388009927854865440685833157284740266993754666100686983963668000'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6952a1}[Curve in LMFDB]+  '6958':+    1225, 1272187:+      number: '6.875747431244947167350986192959672853037098070111872829750537996035696828906747452452615173417982727'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6958c1}[Curve in LMFDB]+  '6970':+    11049, -1139157:+      number: '7.306938398681951117057244041765645004609583842957128507048376307869339878384562360035570730417204692'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6970b1}[Curve in LMFDB]+    63049, 14216507:+      number: '5.473069136981874161979418166688320746109127879506914896224036578567232696329257970494202306589889871'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6970c2}[Curve in LMFDB]+    292129, -157786417:+      number: '10.31062959221811896509479980093023736500091483629358770237883595805539027456447150738643187038741864'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6970f1}[Curve in LMFDB]+  '6976':+    25152, -3995136:+      number: '8.999695749758653079684159313542577698142812454594770630211144883298143885623646973611512041928439199'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6976f1}[Curve in LMFDB]+    64, 111104:+      number: '7.689313970880160944474717650579503952495623405567546826624375891184320417551900563546982043479391105'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6976g1}[Curve in LMFDB]+  '6984':+    -576, -31104:+      number: '9.402764611226386201810235958115282093858784780049702484868095582235901097874764202680853330042502190'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6984a1}[Curve in LMFDB]+  '6986':+    249, -13581:+      number: '7.592898710766729859243989721958631848876441879078958311127966862832992606166759688319734206625170946'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6986a1}[Curve in LMFDB]+  '6987':+    145, -31321:+      number: '7.674868873117435925778007035580581103210975623302795626081558132337706302364390212248002936652949933'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/6987a1}[Curve in LMFDB]
History of Special $L$-value of elliptic curves over $\mathbb{Q}$ of rank $2$

The first version, 2021-05-08 13:57
