curve #4252

y² = x³ + x² − 374432105959282528923780092529115093500321x + 86223539005623480721627083722362048027214469631541079793514079

a-invariants: [0, 1, 0, -374432105959282528923780092529115093500321, 86223539005623480721627083722362048027214469631541079793514079]
c-invariants: [17972741086045561388341444441397524488015424, -74497137700858687343593636782637082863843350428299878088524256768]
rank (lower bound): ≥ 14
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 1023842306084702920997070010643660906803509879520277722560
discriminant (Δ): 147986640262703374388318698088482713109313749450364614434892561248043811278017683166068965199845885505425316157779210929766400
Faltings height: 22.8004
naive height: 298.7923
primes of bad reduction: 2, 3, 5, 7, 11, 13, 23, 37, 41, 59, 71, 103, 191, 227, 241, 277, 293, 373, 641, 677, 683, 739, 769, 4969, 9623, 27779
regulator: 2130328475931343781.17115154909866649823429990558195339203568280904
submitted by: wgxli
submitted at: 2026-10-15 18:27:29 UTC
last updated: 2026-10-15 18:27:29 UTC

Witness: 14 independent points log in to add more points →

(-192687496455694750590846/361, 79402266142690250051058059073337095/6859)
(153020631354564045710271/529, 17705364445301438557007005816088160/12167)
(33192295655553050335531917279, 6047222321167184471443971716274207603919200)
(1140668754500984200779186/2209, 573976088209655623468242390334479675/103823)
(1328419208269470336277386/4489, 347384482588875732434910408330378435/300763)
(3389157738157552530339831/1369, 6063999183472436528047698556794656160/50653)
(57887878288688155894145205/187489, 18043041512369571061376032535803559964/81182737)
(76983547048669764307012650/187489, 106040137432526389886178410716468517649/81182737)
(110253291541052582422538061/156025, 814090693739676309213081270865426635516/61629875)
(145887427570399162859843839/136161, 1519825352128691209812536671156941125920/50243409)
(-448674092990159281813190019/1247689, 18403214141564084784594929394755989824300/1393668613)
(2556332422902517624359883431/10156969, 91144603388609125046687785114600592819040/32370260203)
(477667975308496851928599433977/979752601, 135626325899059197487009882436514993148423032/30667236163901)
(961256712598207972771831726885650/27852605881, 29798284414000255951861839752453894029415157337017/4648349248085971)

Commentary

specialization at T = 23695/2238 of the rank-5 fibration of X400 given by: y^2 = x(x + (121 - 99T)(26T - 23)(75T - 44))(x + (21 - 74T)(25T - 36)(T + 77)T).

last edited by wgxli at 2026-10-15 18:27:29 UTC · history