curve #4187

y² + xy + y = x³ + x² − 45832223968854421589885130376226191745x + 119302226161484596742096486192775906838936727112415905407

a-invariants: [1, 1, 1, -45832223968854421589885130376226191745, 119302226161484596742096486192775906838936727112415905407]
c-invariants: [2199946750505012236314486258058857203761, -103077123403522691601670964699345975281199979160568771300009]
rank (lower bound): ≥ 25
torsion subgroup: trivial
conductor (N): 23245678877229473722476726532968739564574826882685018334849453662724503354377590572824690 ★ record for rank ≥ 25
discriminant (Δ): 12924460012696074420095146791118469958548390890119320909042371598602360653011015688790990825180995190172160000000
Faltings height: 20.4225
naive height: 271.7678
primes of bad reduction: 2, 3, 5, 7, 11, 19, 23, 29, 79, 131, 7998297600445324621, 872091769630260208978461227089207530645161142854134055377
regulator: 10484040136577878626124740.96824913182185286289286047218641691479598118493762814546447605
submitted by: apkorte
submitted at: 2026-10-11 23:33:53 UTC
last updated: 2026-10-11 23:33:53 UTC

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

(4849152048724378047, -3328528365654842802704537024)
(-620327327803177153, 12144731430424878176911331776)
(126143163082372444047, 1414758001189315967845145106976)
(3804186060982898047, -38191543753182994952881024)
(2546033700748515247, 4372180468205076680387918576)
(3433025740283185547, -1555479146957666879326893524)
(5495786360368065924093/1369, -4289449832264122329120183108422/50653)
(3546515776226775935, -1168244519334703932664947584)
(794694589045453962279/25, 21931278458184478075218828808448/125)
(4076474056291704447, 457968915598407006339381376)
(-102141767600716716186473/48841, -154922994667822050893574927262693664/10793861)
(-4153089581755554753, -15427716394030408235548591424)
(-7615761992743460253, 5161068599207629302218059576)
(4053633543656633547, -352517746940309402336372024)
(4093945363526008447, -532654398283642392649594624)
(3416544009286592927, 1610887480991751273086181056)
(4012303473974275797, -42659469353683898963561774)
(5402531835896069247, 5420126003709845059557890176)
(4556436219606167661, -2251066209812062633631208566)
(22211849577519808215, -100298858190310380200630350472)
(2831626321792881377, 3496681137962638512873350366)
(-4498562880368581953, 15311552659356456077526078976)
(3768320657057186687, -320577157163222399689416704)
(4310872334417932047, -1355350282864435030755839024)
(1211455858138450076863/289, 4498074710032854938742610946368/4913)

Commentary

Specialization at T = 1815/788 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -28627871103769984370352T^8 + 250727503810391437739328T^7 - 288854657346414564166464T^6 - 1046563569239575640345856T^5 - 2237723908977579748622280T^4 - 2929539083122814891913864T^3 - 75441235775660299609524T^2 + 875991646929384189258912T - 320782968880211172761712, A6 = 1174522160214039703024876049652096T^12 - 16131442871155893431215736050050816T^11 + 93303642018533859944970389927921664T^10 - 231262072230896017869661906694261760T^9 - 366595866204221333220660871885856160T^8 + 1809536503795113401754131369092034592T^7 + 3099112034539070371703903967203738928T^6 + 554652868729364853437329955292672288T^5 - 394633275566228428167096479633610540T^4 + 857804045527423469286999564571793520T^3 + 199633165297674561155987038442283696T^2 - 284106783901779659207593408557902976T + 70103120599376154084348899681491584. 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 2026-10-11 23:33:53 UTC · history