curve #3386

y² + xy = x³ − 495343314x + 4240373022468

a-invariants: [1, 0, 0, -495343314, 4240373022468]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/8ℤ
conductor (N): 6618066
discriminant (Δ): 10713335734636867166994432
Faltings height: 3.7330
naive height: 71.6759
primes of bad reduction: 2, 3, 7, 13, 17, 23, 31
regulator: 9.57032739987868529027103429038567741219073
submitted by: David Renshaw
submitted at: 2026-09-26 21:18:45 UTC
last updated: 2026-09-26 21:18:45 UTC

Witness: 2 independent points log in to add more points →

(-4284, 2508894)
(4896, 1387710)

Commentary

Found in a rational-parameter search of the Kubert–Tate family for torsion ℤ/8ℤ: y^2 + (1-c)xy - by = x^3 - bx^2 with b=(2t-1)(t-1), c=b/t, at t = -4/13; converted to the global minimal model. The 2 witness points were found with PARI/GP ellrank. Conductor 6618066, torsion ℤ/8ℤ.

last edited by David Renshaw at 2026-09-26 21:18:45 UTC · history