curve #4234

y² + xy = x³ − 87311260961484204826792078128379275970x + 361111361665657914850995183429023003576489538566289001412

a-invariants: [1, 0, 0, -87311260961484204826792078128379275970, 361111361665657914850995183429023003576489538566289001412]
c-invariants: [4190940526151241831686019750162205246561, -312000216479128438437546249271902737837615990946517005089809]
rank (lower bound): ≥ 24
torsion subgroup: trivial
conductor (N): 58219598583889402267386060059408058903194768838964817551390692790917204922228640375910
discriminant (Δ): -13735259866556888673444576442144357662758639946498113227088464928848250077159178701215696442846163744369869824000000
Faltings height: 20.8173
naive height: 273.9807
primes of bad reduction: 2, 3, 5, 7, 11, 13, 83, 97, 139, 11617, 278491, 4317816702373, 80316743721719, 118776478958706660989128744648406799289
regulator: 3041973519286326786080174.8323011709909848504638720602552896286655746527771314978561530
submitted by: apkorte
submitted at: 2026-10-15 03:24:33 UTC
last updated: 2026-10-15 03:24:33 UTC

Witness: 24 independent points log in to add more points →

(-4693227481740211036, 25836178888676683699093828718)
(37581531059750504204, 223961833825845210489921673958)
(7116346323032784644, -10008143736117182275074210802)
(6937879390984378214, 9450155167987489328409042818)
(6278674593843288014, -7773592165697177684625203632)
(11402158614586750514, -29119757480090846177575378132)
(-10951694666097647266, 1943084969116608638492552738)
(5381326263004597514, -6862728618224852494712784382)
(-5784687579082348636, 25934706127905821560979956718)
(9070440000156921764, -17759815480804799985694804882)
(-911310000510532336, -20974321366767892116515469682)
(24778525165209201764, -115806173157827573850981244882)
(39695715021529387331/4, -173803092260204651728843229381/8)
(6794099022214461764, -9029071686268296904144024882)
(-6182923758107481436, 25779566424691599622349639918)
(66784699504235272664, -540742924622746314677523729682)
(-513899506230609023164/49, -3807658561951858761956743012126/343)
(4205005179848064164, 8265606264802432300248993518)
(-8845907921848133836, -21006324579629607812579894482)
(-1046391139169601557752/841, -527513170772513988811388058509006/24389)
(-677556879794204956, -20492892722459646878493634642)
(3506518875130245044, 9902918367901619468179789598)
(157450003582410538984964/32761, 42965190208478117168733057582683798/5929741)
(13080938904552169028, 38174469725741615838927509966)

Commentary

Specialization at T = -1072/239 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -9326637192413076012400T^8 - 34884996069313200087200T^7 + 110925288656556862233440T^6 + 385813899399942596207360T^5 + 233890070443452935336560T^4 - 596691564815170847368240T^3 - 9841086512215258337654200T^2 - 7147300944594901868231960T - 1372341018983665060065307, A6 = 355645975300077069151201411600000T^12 + 2273247912229167049862154805705600T^11 - 7137338446733984479207110102193984T^10 - 57261816964678192543182240688567360T^9 + 101108657979008556764293700373699600T^8 + 692858910748462408271944263396908640T^7 - 1034224818521729452686537198181505760T^6 - 4446809343349652848812523474435426752T^5 + 4366238810355442164044492645139162240T^4 + 13060804253335292158409237965744231280T^3 + 12512018819180205106449295745222847800T^2 + 4777750191287146089865021403646408520T + 622816136791651503989297338315313206. It is a 2-neighbour of the fibration given in curve#3900: its Mordell-Weil lattice (rank 17, no reducible fibres) is a root-free Kneser 2-neighbour of that fibration's, realised by the pencil |O + P - F| for a section P of height 12. Constructed and searched by Claude (Anthropic): a Mestre-Nagao sieve over the fibration, then a 2-covering point search seeded with its generic sections.

last edited by apkorte at 2026-10-15 03:24:33 UTC · history