curve #4233

y² = x³ + x² − 397993975975448665835891396209594020x + 107503387191349723642262147446214535337504221372561600

a-invariants: [0, 1, 0, -397993975975448665835891396209594020, 107503387191349723642262147446214535337504221372561600]
c-invariants: [19103710846821535960122787018060512976, -92882926533326161341536760474458574292340369374256300224]
rank (lower bound): ≥ 24
torsion subgroup: trivial
conductor (N): 5227522119347253134367698504405002351684899917802529603902390934965733498909951087093080
discriminant (Δ): -957931128596215762462972924449444131268846999209717221848474201804550816440881935376185193710551807708000000
Faltings height: 19.4515
naive height: 257.7419
primes of bad reduction: 2, 3, 5, 13, 23, 31, 43, 79, 89, 994489, 8830452193, 2554614859020113, 48790382850554741353, 1092471856440567451788553
regulator: 12344453430040317632674191.751109999265547187778819639975393075165414805258636155359798
submitted by: apkorte
submitted at: 2026-10-15 02:59:30 UTC
last updated: 2026-10-15 02:59:30 UTC

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

(382274873004612735, -105941856132035394619209750)
(-733762256661954090, 66884778856483122824149050)
(-217075561574575440, 428566403203999761416398800)
(-127482891993540257760/289, -2181896671716575570062094747400/4913)
(-209440004917830540, -426230146091717708928275400)
(56457669740665525920/121, -203532125972948670858927936840/1331)
(-1436865894430989435/4, 3614347872789281464563572625/8)
(235317344242179060, 163948235057282387870098200)
(334926039987568683, -108514150638656934136947696)
(352788686940847110, -104898259032038671348790250)
(565023529184220660, -251022081780753881859586200)
(4377057159736170790/9, 4595971302249639624628251650/27)
(316773329294558448, -114961900189726413641430144)
(-326994002478031950, -450201256383742579972880910)
(1698803943645516660, 2081832638989456422477649800)
(-404342776091064690, -449802323507323780638354450)
(-630350257745993040, 328503742325074800617000400)
(14657163847995806865/64, -86234205952039758455250231975/512)
(1060008195356799810, 936306646391565091687763550)
(430226640028340640, 126129138794453642145241320)
(-591149707063693920, -369046649169741442258592280)
(110995449378270075, 254352766409911677367036260)
(36919739458863247365/4, -223821052069886825357634540975/8)
(12122605872548589435, 42152004916174281232237001700)

Commentary

Specialization at T = 282/31 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -197025213169039645478907T^8 - 341931841726476599360316T^7 + 665947206984249248496984T^6 + 768079108867411834646148T^5 - 206522586961384218063030T^4 + 567964478469951746923788T^3 - 1467887770507529603462256T^2 - 589605913192300591575636T - 60265701752223364133607, A6 = 36494962374013345038748480651016406T^12 + 110908265109200423180608249500271092T^11 - 159142246935958461715703643687536484T^10 - 720270176428343092357105513183159380T^9 + 221065402893902851082492835239899950T^8 + 1953863404097225126588131072937056920T^7 - 202924758161403130921394088084358680T^6 - 2477568108201452732342173930140320280T^5 + 261194239478925852479287144045333650T^4 + 771382032898538394406350724439690100T^3 + 379387733802661108189747796000121180T^2 + 80405723305695990012919388855834220T + 5982031282168624025426587018231050. 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 02:59:30 UTC · history