curve #6388

y² = x³ + x² − 397364674052710462575518446764x + 96375521637447910551227209827899938086962316

a-invariants: [0, 1, 0, -397364674052710462575518446764, 96375521637447910551227209827899938086962316]
c-invariants: [19073504354530102203624885444688, -83268450694755109157286436471918768256448109120]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 4197634793865504992224501919470975981296
discriminant (Δ): 3054801345415847256451965115791412312087805395457008085901580135193140055054688481722624
Faltings height: 15.7458
naive height: 216.0776
primes of bad reduction: 2, 3, 7, 11, 19, 23, 29, 67, 97, 101, 173, 191, 467, 503, 653, 2383, 6581, 22307
regulator: 9151462517.56000249933263144585715028281624253584502087316649
submitted by: The Wren
submitted at: 2026-11-02 01:06:30 UTC
last updated: 2026-11-02 01:06:30 UTC

Witness: 11 independent points log in to add more points →

(1428145737661077/4, 982413753086123269065/8)
(-33396305934017358/121, 18104732597302590979327560/1331)
(444151511353302, 2739237248836895474520)
(-2789115401535375/16, 810420204084240657391281/64)
(902144847547302, 21728309615619535439280)
(1361874658708902, 45618933494107635138960)
(8952140649023846/25, 3025843060336703473944/125)
(20451359061529538, 2923339880171489386128360)
(-136426193682871991691/770884, 8592243858407511299576966032071/676836152)
(8762987347374879/25, 49632195787030978960908/125)
(2569954263485468694/6889, 133676638514328265158526416/571787)

Commentary

Z/2xZ/2 X400a family y^2=x(x+A)(x+B), A=q(121q-99p)(26p-23q)(75p-44q), B=p(21q-74p)(25p-36q)(p+77q), at T=p/q=-44/2001. Family equations as posted by wgxli with curve #4252; section formulas from the Layr Labs Elliptic Rank Challenge repository (submission/research/x400a-family.gp, commit 7482bf9). T selected by our Mestre-Nagao sieve; PARI/GP 2.19 ellrank seeded with the family sections, bounds [11,11]; torsion Z/2xZ/2 (elltors).

last edited by The Wren at 2026-11-02 01:06:30 UTC · history