curve #3393

y² = x³ − x² − 1661024x + 454569984

a-invariants: [0, -1, 0, -1661024, 454569984]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 85068816
discriminant (Δ): 204248718332035743744
Faltings height: 2.5951
naive height: 54.5824
primes of bad reduction: 2, 3, 7, 17, 53, 281
regulator: 36.34652530478295612169338456330656251202086406
submitted by: David Renshaw
submitted at: 2026-09-26 21:19:55 UTC
last updated: 2026-09-26 21:19:55 UTC

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

(-846, 35406)
(3088, 157360)
(34560, 6420288)
(-3680/9, 881216/27)

Commentary

Found in a Mestre–Nagao-sieved search of the family y^2 = x(x+a)(x+b), at a = 833, b = 2529; converted to the global minimal model. The 4 witness points were found with PARI/GP ellrank. Conductor 85068816, torsion ℤ/2ℤ × ℤ/2ℤ.

last edited by David Renshaw at 2026-09-26 21:19:55 UTC · history