curve #6397

y² = x³ + x² − 40511147191289311775961x + 452725554469574211419133216252135

a-invariants: [0, 1, 0, -40511147191289311775961, 452725554469574211419133216252135]
c-invariants: [1944535065181886965246144, -391154879073379329057222420633321472]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 2390681677049094299819723099307840
discriminant (Δ): 4166496624233158840971977128041063799434250446329887599722885304422400
Faltings height: 12.0487
naive height: 167.7812
primes of bad reduction: 2, 3, 5, 11, 13, 17, 29, 31, 59, 83, 103, 113, 149, 163, 173, 463, 2437, 4217
regulator: 75578663.16930944786094799939314904176017549212337387099308
submitted by: The Wren
submitted at: 2026-11-02 01:17:50 UTC
last updated: 2026-11-02 01:17:50 UTC

Witness: 10 independent points log in to add more points →

(621677286327, -464257848110612160)
(-176827028913, 45685878583380360)
(-152260488966, 55597469941290339)
(301427054277, -125014839870272040)
(-31305181503, 41112709006673700)
(532308942867, 360165954412426800)
(1079613042987, -1102304317222556760)
(248562725409, 75764251220694036)
(-191642719156653/961, 730243438563114113040/29791)
(732092990947/81, 6812021401736559040/729)

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=-110/59. 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 [10,10]; torsion Z/2xZ/2 (elltors).

last edited by The Wren at 2026-11-02 01:17:50 UTC · history