curve #3359

y² = x³ − 67921x + 5635680

a-invariants: [0, 0, 0, -67921, 5635680]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 318317792
discriminant (Δ): 6332888543984704
Faltings height: 1.7507
naive height: 44.9919
primes of bad reduction: 2, 13, 17, 19, 23, 103
regulator: 80.74771508933936142790945847572886098712221367
submitted by: David Renshaw
submitted at: 2026-09-26 21:11:45 UTC
last updated: 2026-09-26 21:11:45 UTC

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

(-48, 2964)
(28, 1938)
(47, 1596)
(1473, 55692)

Commentary

Found in a Mestre–Nagao-sieved search of the family y^2 = x(x+a)(x+b), at a = 103, b = 494; converted to the global minimal model. The 4 witness points were found with PARI/GP ellrank. Conductor 318317792, torsion ℤ/2ℤ × ℤ/2ℤ.

last edited by David Renshaw at 2026-09-26 21:11:45 UTC · history