curve #3593

y² + xy + y = x³ − x² − 9526711476845901608280382793633750919674617782992759276238362125085687x + 463206907181874406177037011231363730963974319416722261144137263207860246343963112239868204998888573429711

a-invariants: [1, -1, 1, -9526711476845901608280382793633750919674617782992759276238362125085687, 463206907181874406177037011231363730963974319416722261144137263207860246343963112239868204998888573429711]
rank (lower bound): ≥ 31
torsion subgroup: trivial
conductor (N): 7196041312705198647586769585657137954320880499941576604522603959364046714544035788021082339260729646489743926399544179737919664755535589500866043636248384367450976255349850264370
discriminant (Δ): -37354036867943778396997554476006589814870747758500939720300541060758570729525851690733093957516022836768756354644354125834730089825437182943216497537639246618853190116975756014347912884563650048404602492800000000
Faltings height: 39.3061
naive height: 495.5269
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 23, 37, 41, 43, 103, 1583, 2143, 93625397, 5980229023, 1907622924307, 21197931858187027287557, 229389262935827220620886349946263120571, 998114498876752180590165653101118737687296085424865606677704189
regulator: 273954937096341441703707270798789075836265.1528687623357817279683253434039275556527315078756670296060
submitted by: wgxli
submitted at: 2026-09-29 06:05:57 UTC
last updated: 2026-09-29 06:05:57 UTC

Witness: 31 independent points log in to add more points →

(-5515667988066484253173413839555939, 22706503116551854837813714112857474195848110785354424)
(-8218642219266668233124346553712909, 23258297514478733987652196704312237777278940532264094)
(25289423452240850802869647797004931, 15442015483688527075171854476181793675300351449735134)
(38716273931629572162501164676956851, 12345116023878787787255119222969018673219084297796254)
(42305885614897402215272083264293931, 11657165425952370383680271086221100280508727133818934)
(43707992556214663983330458898915331, 11415459874925119600034525818234092722792043797482334)
(45731835230733101963308479945958841, 11098490991831278753475890900826956524388252726220314)
(51057693786559570515463430068252579, 10483161922073231750498313898043999653629464654222366)
(56393610268470496280017189627006981, 10261892003095398267299750661776959858763636985635984)
(70883114285985610976094371969812741, 12002923586050681529283280307784252814827471497329514)
(79891156588326866423153990405567831, 14560905098000309245783311110341940241685754351032334)
(-116184108806706709445017575498635309, 1311290815113076889895827869301995608939953847703694)
(-18429680328525187550268629638390919, 25149979242797099932782147356873591000215051843319834)
(-21361017094012830310135935587443709, 25631236221064481321591380134628305805807291771616414)
(-70689507279912309466046658317240189, 27989453806690024712241445732444460453347379873054174)
(-72741379357992255459135647064744941, 27772215945426442515455255237619193512793027560415646)
(-74975178358295287215864581270940269, 27495772873659102965063716520435295381300668479331534)
(-88074283347075888403588466824890749, 24881036659957400049042216561130353635347255392242654)
(123451906689225929652705988467655331, 34184343625879673495396836887986027256682724614882334)
(133563315004465937886789964877715331, 39666673285421871487429278798580404726900701889682334)
(-106708078214660288865811164181460339, 16270841648060735427936928851727363243342883588945864)
(1223123260506047969392822242017681809, 1348569827702526602804528576200080917897363703890697044)
(1359213992068263529420393404185503651, 1580699682650308637657589011690857742468360045693441054)
(1446574675512500253654447661620725731, 1736016379772457940064166353385117978463098798271169534)
(2603271632076432593054874316354280491/25, 3062622089319064085879178898779231707718566886203908774/125)
(76122783668470924595879384538262356211/361, 606951603391216789561714176241442416052643178223833471946/6859)
(4315294783266558879983089266710853965459/3364, 282685034989457238853037027057512404588714211201593896101433/195112)
(1606548540810930830673559226858238491251/19321, 42118674501487726875810506591489829579860196767751846426746/2685619)
(184343405324842213410210354878411287756639/1369, 79148245957611724094225582579945957360588240035299803007086102/50653)
(783331738456205104150160176465694385571/112225, 749130111890695641847680402379911134083609511065173337704514/37595375)
(217163153667942248938850322849706195948821859/2749219489, 2056606691694398307231045743043694088700397399517248253707095188158/144149825466737)

Commentary

Specialization at T = 1724318/357675 of the rank-17 fibration of X1152 given by: y^2 + (Lx+B)y = x(x+D)(x+E), where B = pE + qD + pq(L-p-q), L = 264419T^2 - 504384T + 75701, p = -95942T^2 - 182785T + 5472, q = 8310T^2 - 15587T + 271053, D = 7007510972T^4 - 10559535372T^3 - 54483507557T^2 - 33779008458T + 719232840, E = 966(2050170T^4 + 30830408T^3 - 119670167T^2 - 13513983T - 1132428).

last edited by wgxli at 2026-10-04 18:08:14 UTC · history