curve #6415

y² = x³ − 32934248383899316x − 656151816872537568467040

a-invariants: [0, 0, 0, -32934248383899316, -656151816872537568467040]
c-invariants: [1580843922427167168, 566915169777872459155522560]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 1238609546663682784153024
discriminant (Δ): 2100256290839300465334827427291669581795735183724544
Faltings height: 8.5392
naive height: 125.7135
primes of bad reduction: 2, 7, 11, 17, 19, 29, 37, 47, 71, 79, 109, 769, 887
regulator: 1043329.52441410496990458165171866330314317390801195317
submitted by: The Wren
submitted at: 2026-11-02 14:10:53 UTC
last updated: 2026-11-02 14:10:53 UTC

Witness: 8 independent points log in to add more points →

(-38066747, 736468755483)
(648970717, 15852221736351)
(-1039636508/9, 34225601204564/27)
(211154613, -1343214957843)
(-20309448, -65935387944)
(-68022428, 1126664323884)
(1505098682, 57959535668496)
(-3091022886108/18769, -1410848431336194444/2571353)

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=-374/37. 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 [8,8]; torsion Z/2xZ/2 (elltors).

last edited by The Wren at 2026-11-02 14:10:53 UTC · history