curve #162

y² = x³ + x² − 1029869656x + 15913087721444

a-invariants: [0, 1, 0, -1029869656, 15913087721444]
rank (lower bound): ≥ 9
torsion subgroup: trivial
conductor (N): 1312277573656056405360
naive height: 74.3195
Faltings height: 4.1996
discriminant (Δ): -39490526497340544133267123200
primes of bad reduction: 2, 3, 5, 43, 599, 115831, 1832712667
regulator: 1725985.6630133732536627998069706320686406126996309353094
submitted by: Seewoo Lee
submitted at: 2026-06-25 06:08:44 UTC
last updated: 2026-07-01 22:54:24 UTC

Witness: 9 independent points

(30800, -3662262)
(104384, -32338710)
(5201141/4, -11858161785/8)
(72821/4, 14306745/8)
(122216/25, 414489186/125)
(1528496/49, 1289868390/343)
(1109966/25, 949593564/125)
(3929870/169, 4680903876/2197)
(10635650/529, -22218226104/12167)

Commentary

Use Fermigier-Mestre's quartic parametrization with (a_1, \dots, a_6) = (0,6,12,14,15,23) and t = 1. This 6-tuple may not come from specialization of actual bivariate polynomial 6-tuples of Fermigier-Mestre.

last edited by Seewoo Lee at 2026-06-25 06:11:29 UTC · history