curve #6197

y² + xy = x³ − x² − 56243511x + 157223111913

a-invariants: [1, -1, 0, -56243511, 157223111913]
c-invariants: [2699688537, -135828620094429]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/3ℤ
conductor (N): 44994546
discriminant (Δ): 709939266885177481162404
Faltings height: 3.3466
naive height: 65.1492
primes of bad reduction: 2, 3, 17, 19, 71, 109
regulator: 13.03070158699445415929240009974197225444162543
submitted by: Andrii Prokhorenkov
submitted at: 2026-10-30 15:41:22 UTC
last updated: 2026-10-30 15:41:22 UTC

Witness: 4 independent points log in to add more points →

(-2238, 522543)
(1911/4, 2887647/8)
(5004, 30087)
(-38223/64, 223680843/512)

Commentary

Specialization at T = -40/17 of the elliptic fibration of X648 given by: y^2 + (Lx+B)y = x^3, where L = -152T^2 + 280T - 200, B = 432000T^5 - 1539000T^4 + 1782000T^3 - 675000T^2.

last edited by Andrii Prokhorenkov at 2026-10-30 15:41:22 UTC · history