curve #4196

y² + xy + y = x³ − x² − 1636204316577291435600184386647088960758937x + 1156791606392160788311565921568969479338897882099581553779552249

a-invariants

[1, -1, 1, -1636204316577291435600184386647088960758937, 1156791606392160788311565921568969479338897882099581553779552249]

c-invariants

[78537807195709988908808850559060270116428961, -999467947922826921100839536103208935198718130306522691250009213041]

rank (lower bound)

≥ 27

torsion subgroup

trivial

conductor (N)

2354916239112007892073152869234207119051588829446921311745649592265246092225042640809243177031557188527240414952810

discriminant (Δ)

-297743220529971557422610830343358680816070984996754776293483102932642726894292413999595938037364363301326873960258312657152000000

Faltings height

23.3537

naive height

303.9402

primes of bad reduction

2, 5, 7, 11, 17, 6089, 45589, 88431538658940336339975361639116868405067, 666238784557858910525818998986943572963543744552344842686417

regulator

6931090223496514928578101729726.3202399948415484023659143849970968015863772112105095849527082

submitted by

apkorte

submitted at

2026-10-12 04:52:15 UTC

later contributions

primes of bad reduction recorded · Daniel Błażewicz · 2026-10-12 05:04:39 UTC

Witness: 27 independent points log in to add more points →

(1025354926528389774507, -23603202030014569915902684227704)
(1130820936257276474477, 27433219914898873424338642197026)
(243117974671910591501207, 119872513000974118828179606042657196)
(1036836872431809445879, -23978033884145110898784281245908)
(-994893780474147764713, -42425012383953089098408291684804)
(846993001618073234357, 19456833299263906081765728223696)
(-560672230776643928753, -43565086208439559080504903909404)
(438911095408149723013607, 290778972884555875725411562562449196)
(4226408575616889480327, 264075503013194427324661056337676)
(5408986642763711690607, -388018098621624591070028878938804)
(4687396896923564933607, -310607924154523149115686674542804)
(122184542682401365902583/169, 41204730135484100920286453294031612/2197)
(781160994280046245517, 18850114536085176350158205760496)
(11955852221994028703833/9, 983747803908506954614902566084432/27)
(-101722819790767765193, 36361770351459512347909639089196)
(-2088897983340399110220337/1369, -508667413342621650603351597557716612/50653)
(19569270897037291414637/16, -2008842429432523926702663257730781/64)
(870165267558112173607, -19796509385034880897769140902804)
(47827993433848136492297967/3481, 329405342026900493186686330527718013284/205379)
(2625516714328936065903/4, 152967736762800852061665463574943/8)
(3120117263748949907151963/2809, 3966385721092977515658355892654188192/148877)
(3540791058480848407764127/361, -6609918563989598910339647893428728636/6859)
(-107683085987849143813, -36492937418767493576570942880884)
(10894375924188849344165845561543/18661472449, 51021048931240204527627884088089920007355469428/2549287766840543)
(-100619538762485959412193/121, 58660323476416373544527204685242676/1331)
(611566000835364676833103/729, -380937767694049945481313064729043132/19683)
(873283662534589760887, 19847128211153255150718993005916)

Commentary

Specialization at T = -257/534 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -4247232044073407555787T^8 + 31562440352380138911048T^7 - 1079699554459262036768544T^6 + 808673010384011405337504T^5 - 370492253920133220842280T^4 - 583074314969616668304144T^3 + 1749298418924942437857936T^2 + 6981774061823935535232T - 733152591313571732712192, A6 = 99353320243156880097765275026566T^12 - 1192047168420920079142624551372696T^11 + 51816962014333011197785218484470384T^10 - 160733095313938224161654547702105120T^9 + 1161258380951665950605765814917343480T^8 - 500375410396179478662655437513305328T^7 - 2173656091315178125605895871772822192T^6 + 963181788398399075637606795285979968T^5 + 1772421005485324185894755005716179280T^4 - 460279153080260982936199267860712320T^3 - 927109903404780261014801473928185344T^2 + 58145438007481575316035550439430144T + 241998303936746269490887631501697024. 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 2026-10-12 04:52:15 UTC · history