curve #3717

y² + xy + y = x³ − 381330508249700150893998x + 90635915255018855118215059707148256

a-invariants: [1, 0, 1, -381330508249700150893998, 90635915255018855118215059707148256]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/6ℤ
conductor (N): 447060738391164495290
discriminant (Δ): 1917523281786781390971332623201098397040724314837167577694272000000
Faltings height: 12.0804
naive height: 174.5075
primes of bad reduction: 2, 5, 7, 11, 31, 41, 43, 59, 61, 73, 109, 409, 907
regulator: 19096556.50190866289388076085736561219199829862410368
submitted by: Michael Rubinstein
submitted at: 2026-09-30 01:49:04 UTC
last updated: 2026-09-30 01:49:04 UTC

Witness: 7 independent points log in to add more points →

(142212243658061/400, 8127223157255448899/8000)
(2046212739609181/5625, 3171397777536389808071/421875)
(3117555472318266/9025, 9780986299017744216476/857375)
(88911667482176845/249001, 67731950524310263539423/124251499)
(103907538642461605/247009, 8383639402834992869533421/122763473)
(1707255230699357645/4791721, 1911699047366521595207713/10489077269)
(6314317910825566645/17631601, 122030791666362883136340923/74035092599)

Commentary

Specialization at t = -2076829/55327 of the universal curve with a 6-torsion point, y^2 + txy + (t+2)y = x^3 (as searched in arXiv:2003.00077, Sec. 13), found by searching small-height t = a/b for small conductor, with points from a 2-descent (PARI ellrank).

last edited by Michael Rubinstein at 2026-09-30 01:49:04 UTC · history