curve #4043

y² + xy = x³ − 8570253126523615890841680541683706593823098045x + 303027927833848473426060414809108967878826721278349547510378356583025

a-invariants: [1, 0, 0, -8570253126523615890841680541683706593823098045, 303027927833848473426060414809108967878826721278349547510378356583025]
c-invariants: [411372150073133562760400666000817916503508706161, -261816129648445081040116815453295257947650427785493010275841655350792841]
rank (lower bound): ≥ 27
torsion subgroup: trivial
conductor (N): 26053726879869645816679754090304616360346051455724068011479041708711444194276344902246517917083676530 ★ record for rank ≥ 27
discriminant (Δ): 617828680631258882418623580542062286979421149238747477413157834867141249199092299675260695001991210154654775918118349403892286329600000000
Faltings height: 25.2629
naive height: 328.9075
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 37, 43, 47, 53, 59, 109, 467, 888761, 954469099, 2164829083378539459792001757, 2334833037255101842475964919405642227953
regulator: 4249393195658043334600052335.0818549699978605244788686178115667064147111134255958843319396517
submitted by: Matej Jurasić
submitted at: 2026-10-07 13:36:48 UTC
last updated: 2026-10-07 13:36:48 UTC

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

(-22274517573165776316510, -21974407563105512292771395366765745)
(58819117772520117614490, 1558535202684937061318842461552255)
(-16226186884819770142110, -20924107577409014307128880257549745)
(57523648739997578940690, 616226254496037628565376538283055)
(38645591515855350977202, 5435247162593217284881385538040287)
(45861105739296902802090, 2538425059404992743967346763178655)
(7475502035553589687402690/81, 12587799571230519474346269979317211895/729)
(69911526519375183335490, 6750520871004636671757406860282255)
(468421009778461795085490, 314751737046524063724778787694732255)
(-30555264061895574353310, -23159601516221623842314791872653745)
(57258948360054356917890, -180831236185332302937101101568145)
(57773432067719094768930, 854042018459370843019363389302415)
(-150652248972200519059821/4, 191396487375654987637699640051191311/8)
(47755676459228472136290, -1631413309509186587529473308781745)
(-104841799591411177476510, 7010583957512409600330374604594255)
(48908896669568741982090, -927526422554344186114062815226345)
(101142085079526653724450, -21699539316141274805279526939035505)
(34905584855919951661740, -6812282259743187574850041326497745)
(219897165124926639380490, -95139523414239708734892505175747745)
(21169232398899302910990, 11449408337832653014992442767507255)
(1799000729722648471362, -16959241094395681028997556274600049)
(39201714147429417906210, 5225275593173885807532288762126735)
(245329154544315256351965/16, 847183234852984863816493297674069945/64)
(75140777501825260992270, 9127295698652016892275593204337975)
(501386896168931630459490, 349355227997250892786466382424338255)
(26887685983691809968728106/625, -58364618332446790016885474641981036329/15625)
(83069864214707454310491090/1369, -128249206795131360622816155559007291085/50653)

Commentary

Specialization at T = -19183/2817 of a rank-17 fibration of X1152 (one neighbour step from the one in curve#3856).

last edited by Matej Jurasić at 2026-10-07 13:36:48 UTC · history