curve #6140

y² + y = x³ − x² − 1569690942057842258857349751202262286896520x + 961024971531151492705937931945503717653222077921354824325657512

a-invariants

[0, -1, 1, -1569690942057842258857349751202262286896520, 961024971531151492705937931945503717653222077921354824325657512]

c-invariants

[75345165218776428425152788057708589771032976, -830325575402914889697478302209602553481832958595704316678741892760]

rank (lower bound)

≥ 27

torsion subgroup

trivial

conductor (N)

15708605030361218183151622290662540842542745818741401673210563958919099010827642867596485832561710060137579851363

discriminant (Δ)

-151454890449840161014632034867280134818209599017201332261277019579712177283671183521159677587108499998902486701270170321317363083

Faltings height

23.3065

naive height

303.5694

primes of bad reduction

3, 7, 11, 13, 19, 43, 59, 61, 71, 276170231, 2275570421931127, 5419285694002261, 155674815484657907, 6751409434747429522431628633386029290934729

regulator

1942392989494641120639052324458.9962281182461462089830556673311957070434284517082295430107368

submitted by

Ritik Jain

submitted at

2026-10-29 13:47:38 UTC

later contributions

primes of bad reduction recorded · Daniel Błażewicz · 2026-10-29 13:55:41 UTC

Witness: 27 independent points log in to add more points →

(3899655328768623042166837566178629/10870354068484, -755258724134356382685592397169551154459433764110057/35839796511581254648)
(142584608056487727010503/169, -33839870924917761805385850710974526/2197)
(-377567074821119429042074083400750188833753264402039/348899109724014370231037551849, 7690049377621489510383797158786439259749430065691486598518250762788351812917/206086617825395714549627922036192135949039243)
(558531855319744331016075125/111556, -12832068473346297512226328141776680333191/37259704)
(509761667379799624998, -17126640522514246655521283878685)
(-4669877277851369290469/169, 69627162904981851404663551588084693/2197)
(-381785176021169710004163/289, -132667591216559861540020686999601300/4913)
(-857093785320220421324, 40948356079626496265348999067256)
(-1127686721616145626612560/961, -1026451265245181141661186223846686839/29791)
(1907984658116659769855, 70085066131989905177148475629621)
(-95155204988138942471465354/64009, -49165894354042112837449766376033722612/16194277)
(486778619643911526477/4, 222250917204480569631533497707227/8)
(1704657950295375723929758217/246016, -69324279675318897503638504474762923318373/122023936)
(-166280184013521730308702/121, 30437516658680463230616542184312164/1331)
(-91515237169675032320919/256, -157388589641166755156093507732541749/4096)
(3778926740985210163284, -221344612526088776667318118505081)
(71344724794436316488920624953717214/131885907937281, 24896230676687902436321556715406749095009897672920420/1514598736611097031679)
(1128643815733535295286019733/662596, 30650785978156650167521701351389891651935/539353144)
(-15766470633564232611603483/21316, -128971559876462944713462981087778515943/3112136)
(1916188953153416970382277102023865508313517234240063299661847236/2194772455677000707824468601154114426293761, -52031500414434176260567796800819103545715573748968818477071015353107211414257245659731612513671/3251503694173740474389272458865958012136962208091732189861333759)
(5773297743023634889432102048538215926825387701791379757254903846658248916010317560230742173829/11442523712541621764615646624513406140623812708174332150380984452586028100, 667596124377962693453822210602445510312700884231757208899192042287684291166680484898444116442039012386478123153779487776527973330722268957367/38706395655518031739229172610739346811560371622338130246811518436900435497261415163834642826060340133551571000)
(95110261646206260038489243186879133522337804935117796057409021378171061314786694417624/199127620232711285552512958564998421180597894835240984943354516441, 1590166776096645416097283949093473972029356377999093990054063763215565098351057424648814534628830839159770651653961169134031134309/88858147582981082761875435070751957010601472214109385078769389058999871538147648468511522145607011)
(-2681547725400565953056379366483617880462510132335988819423775938964183811965641049954608212898170663732/7968523991278043513042209811838652946631044286834623145409835644456281355898092561, -27097062158102274511595105227254088358087766430004255809582016570472363786685390912064799367896151632419530526952061135532153838973342813862987493973252879/711322959618113665963335465487556881812052079233842458851479809233390708444910306961180966282091679554331586879796588519559)
(-203140350570275488518822498494581956128940177399323338095744158775814657467916608813254/237977735258571454810638354488460354657282012515350144634077747681, 4756886435616623906560468817648436619206153106730474060396444764760548966534629927438941603191919477022651091599582848365466176244/116092587454547187287903332412924903435587022581788243445324855729389538393155041404869574665052079)
(2604041082268219473259248363696244846566734430761163661559957094630520349071063154587974545814520094936/5424868806188649534838435546156797315615116864412895637686825381814284851144385849, -7126869044428250410814348761659749162073001336218218904380940320831863719740547305955623640721490153683204144259201338916942795322827569189397217595721035/399561706820770702204760931895174528261286127559105014406578336628420677995809472791132637034122768213640022538823304996243)
(722193167184887958120253023436326039980765846937961961245177058610206248889589035459075844676921522532029/1390607569573990584183870850371713147695979495690477550110635143774411579821531655249, -27727594923571809176567420431616654852263519334799996623178932046412284939494739677284255168765558534052662920047439530528210529821640739754215257952016929828/1639860419384248422990176702534645958179071590613910175829020763195187552695923205665125344761766296090472909540424172536100057)
(-14824711758018668168856576439400921329045362321771100872626687138215831445100864371040115564673596504638072/9989026933836095933270704148784143225406736390394294671524462815269419998841499401841, 147392575545826970441491651896121126780828436264591570542661664310175952447691913297594716894523872423479894630528878284885551879048540876525971870877156295375/31570741059959504302994981604191414863850378084475162962618045816426246875695434166547487173261489792131443043818163648757188361)

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