curve #6139

y² + xy + y = x³ − x² − 206839761068690612365122202090855714692005x + 33642545432374237696622014654889042979130233978121007181973597

a-invariants

[1, -1, 1, -206839761068690612365122202090855714692005, 33642545432374237696622014654889042979130233978121007181973597]

c-invariants

[9928308531297149393525865700361074305216225, -29067159253571341369836743273433295961697655761444925370851715025]

rank (lower bound)

≥ 27

torsion subgroup

trivial

conductor (N)

680723183966080089712488866140619946438518939913458825213086374790472831640631289157482994486102717450

discriminant (Δ)

77399672365136943702124145969372489836809326379089442930924669859194846095477608256221094877234546480807692934860494970880000

Faltings height

22.7076

naive height

297.0119

primes of bad reduction

2, 3, 5, 7, 11, 13, 23, 29, 31, 41, 5431339, 24985321, 100047928343603791708231, 131296340683597457631769053289899921521703950970555757

regulator

33863273128158737535864629717.089782161555943180474582752864265089410783576171662104306303901

submitted by

Ritik Jain

submitted at

2026-10-29 13:47:34 UTC

later contributions

primes of bad reduction recorded · Daniel Błażewicz · 2026-10-29 13:55:38 UTC

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

(-107929981274202726922655892069104766966850334315/207489219750586634601179664, 2086841642121784092068039442355779485463306074549661540761939304057451/2988775581404601041190361601025238475712)
(2093890141680625817629/16, 190017722976318189193344583740875/64)
(-15638488988119352362097081/39601, 57770827802675533933440190002552190400/7880599)
(171411446085306182659, 1795615236797721280441637233760)
(1569908804520884961961909/361, 1956651129110344824251750786108223050/6859)
(8236058694275158536271/25, 140083214262615614236347885836416/125)
(184092919594081301725, 1343039967007534413956660448308)
(165049274173422760434847/289, -49567097855587872357724135142281760/4913)
(-15761542670099597969681/5041, 2095819810004983323956489040622168960/357911)
(164557218088731392257811/1444, 186493695699828569432043767618423735/54872)
(71946414333738229017329/121, -14633413992954246092785191188609060/1331)
(-181942821272687832601, 8077912310345504614804823871640)
(1276562853259773983131/4, 2953418384891706563027687105845/8)
(163844406770254006399, -2037500646520482257823048771040)
(6837418017479096604439, -564154796680883360564971002180160)
(403336156978428140396255/2209, 145474896606642858055149450875634848/103823)
(2103240263896014657709, 94353291954219404420908110447860)
(2090900406152772547087759/361, 3014358077379998353121873593453469120/6859)
(53833895161279106187235, 12490167278791271375059743383207996)
(706910963652086157751931/2209, 49053355203897790565174413809634780/103823)
(-14131679831261737541, 6046711218006923371282358687360)
(405335249427103283991979/3481, -685377935070839123387284016759855180/205379)
(182249113734001804099, -1414050607942757372003562708640)
(99171872743599119182783, 31230447863606918418574726580364896)
(157572641735676487615, -2227700752121798328014708395552)
(-62635122567710324894793326079458119803212689714665449198024177/220877744673764182832715857953158113921761, -865366738448075581267171718157404407841710527377977277514666637700890980666560119948388387440/103807310390196192741284927134226668668762825772634974666386991)
(-2399133895914262802046384691242919612887207715924153/7565020932593576227410372130969, 170750196727246662243867307658917377596756994360790893801255205897249482182080/20807274713505062448125272580443803167559876547)

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