curve #4266

y² = x³ + x² − 548706613704837392209132328067689980x + 142835149952740783674517071497430292197512309490923600

a-invariants: [0, 1, 0, -548706613704837392209132328067689980, 142835149952740783674517071497430292197512309490923600]
c-invariants: [26337917457832194826038351747249119056, -123409569559168037252810254520772941414880745883652704704]
rank (lower bound): ≥ 25
torsion subgroup: trivial
conductor (N): 83649190332021251463700004192640613044692062259091513678741325837066688815236766833685048440
discriminant (Δ): 1759444451470850072684319011374877786144475600337765331769711640725368876180306924429916745456646917812000000 ★ record for rank ≥ 25
Faltings height: 19.5079 ★ record for rank ≥ 25
naive height: 258.4922 ★ record for rank ≥ 25
primes of bad reduction: 2, 3, 5, 11, 29, 139, 157, 241, 8998992345821347, 650595954117933225616883, 70966449329823133133861281900953128449561
regulator: 151860770353785919699884861.3734098434605305872601080008700546279278480960188578065477136
submitted by: apkorte
submitted at: 2026-10-15 23:17:44 UTC
last updated: 2026-10-15 23:17:44 UTC

Witness: 25 independent points log in to add more points →

(-322612288266463310, -535049140095720417696030150)
(15101961994925738978, 58618678989712271461800699114)
(907836714828363740, -626824263049991300346669900)
(273556730485214915, 114908273991395956396567800)
(181946630301187352090, -2454214010398981472388802794750)
(4405724565203813290, 9123721852952122916335842450)
(294607625691332570, -82170976369137785009752770)
(-313294585885923460, 532908061394753138675199300)
(199624270423481390, 203113243331835429005051550)
(6717774573282858590, 17309529326775121999091604750)
(8201454527851561015, -23394531459613302805642843050)
(124398187208413790, 276590116459375384811489550)
(-703025779009966960, 425585356236238734257865300)
(1789232560467106640, 2211116621489700873125829300)
(-670557120399732010, -457449864682014966319633350)
(15680328286312688510/49, 1346710056133300259165712450/343)
(304605923857901980, 62917844794054052046552660)
(183115357750036355, -220224036015020076678159840)
(-824064662148353581, 188140728491421236396742060)
(-590506366020979229710/841, 10403023673830024705152948375150/24389)
(1079154265398875525/4, 961238366486162419140130095/8)
(-71189382008228890, 426070947162502862745060330)
(-24351529057095010, -395199385901991852532873650)
(527276687781423740, -10429805063143987643420100)
(4252941516679212954770/19321, 485786989200385305513145137345570/2685619)

Commentary

Specialization at T = -153/175 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -6276238998830220029652T^8 - 147117265740737393263056T^7 - 1476608115021273186431436T^6 - 2537588378961323810084232T^5 - 2548792821173728264463880T^4 - 1085007595485857335267392T^3 + 1266527474226087852521664T^2 + 648517828968568984999104T - 394596892599474187722672, A6 = 183805362097981758392518604760804T^12 + 5732624758080384420828430168047168T^11 + 73959871695264874512454632891659784T^10 + 444783553219225017837165953570627520T^9 + 2344794983173561552804045815952068900T^8 + 3952769116759113473247850984388751264T^7 + 1034132030293362061211853414656889168T^6 - 2105709468706045084822083295783288416T^5 - 399200243459621989020610997025954720T^4 + 954359071876325078141973753377053440T^3 - 120506571390369672930568955416602624T^2 - 234913768366312396616401364230326528T + 83823186438658476366371918607332736. 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 23:17:44 UTC · history