curve #4231

y² + xy = x³ − 122311635516550357823534835335580x + 510266314127555249409507227897867382022189742352

a-invariants: [1, 0, 0, -122311635516550357823534835335580, 510266314127555249409507227897867382022189742352]
c-invariants: [5870958504794417175529672096107841, -440870095406207744296252002095383181361680081553889]
rank (lower bound): ≥ 22
torsion subgroup: trivial
conductor (N): 90380223496560423965098889851226608711351320575453620625770728236549747310
discriminant (Δ): 4626539043874319346593179481752922544446502769117791674117871396787367068409272295717697792000000
Faltings height: 17.3348
naive height: 233.2660
primes of bad reduction: 2, 3, 5, 13, 17, 23, 41, 67, 73, 2179, 7963, 32707, 782367097, 2825273783, 417202551611, 434424634862545998787
regulator: 561786777273619703513.5206574155350600119056366180074795829726245215714345875612769
submitted by: apkorte
submitted at: 2026-10-14 19:37:53 UTC
last updated: 2026-10-14 19:37:53 UTC

Witness: 22 independent points log in to add more points →

(2008118142453944, -522252730516739578388572)
(2891059597236744, -425229603819768186484572)
(572448999045573144, -433036722794298858483498972)
(-6147556390599336, -1014816618679682224562652)
(11880658462894704, -856784126513825066591892)
(5629181909297544, -11297509933343049767772)
(-8880734812440936, -946616389701893828748252)
(8540241019356294, 297630751221815301885978)
(5420654642352444, 80839142055463720129428)
(-12520993595766486, -280619370182162001395802)
(-8654375966403816, 959478929948960415921348)
(17841496554718254, 2001831907096151916379488)
(20949118731496944, 2672413204283040428507628)
(8292868135044264, -257422561462507045189212)
(140657036367128634/25, -1797607041813274189071762/125)
(-551046235522536, 759933125078994524456388)
(-429221723440356, 750124022660156275163028)
(-323194190332900344/49, -348117440868831067621961796/343)
(-1732211372712456, -846722103806624992607772)
(-318578087513608776/121, -1200888483627883398642066132/1331)
(-11258015062782006, 678512375685649647996678)
(79995498491672601444/10201, -188162215190407269058098481272/1030301)

Commentary

Specialization at T = 50/61 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -10669947921935414766592T^8 + 11909462495223346810112T^7 + 29290850738695015599056T^6 + 749826724411797003953776T^5 - 2800296645908250177432680T^4 + 5291011042236720751525024T^3 - 9053743075112346775820704T^2 - 1373686135035664689321352T - 153062929487822447776267, A6 = 424084459520579325699737587892224T^12 - 702971076900016054991835746922496T^11 - 1428460164091780099364248773693184T^10 - 45531551300070380060674898437602560T^9 + 178145138683306119166957105019549520T^8 - 224307849011465835658368095576866752T^7 + 1890363249575504635685922631864051728T^6 - 8402714629932209203714457518206796848T^5 + 10932033646871752694840038266436614840T^4 - 1045636958253350245306227479236779680T^3 + 3911537622199302231442442603656239344T^2 + 397823992391227625022128206594583384T + 16543056563137220502197778370669126. It is a 2-neighbour of the fibration given in curve#4205 (itself a 2-neighbour of the fibration 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 2026-10-14 19:37:53 UTC · history