curve #3079

y² + y = x³ − x² + 8x − 8

a-invariants: [0, -1, 1, 8, -8]
rank (lower bound): ≥ 3
torsion subgroup: trivial
conductor (N): 40331
discriminant (Δ): -40331
Faltings height: -0.4284
naive height: 17.7242
primes of bad reduction: 31, 1301
regulator: 2.0808883681314856576060184124397381944582699
submitted by: David Renshaw
submitted at: 2026-09-26 20:36:15 UTC
last updated: 2026-09-26 20:36:15 UTC

Witness: 3 independent points log in to add more points →

(1, 0)
(2, 3)
(4, 8)

Commentary

Cremona 40331c1 = LMFDB 40331.a1: https://www.lmfdb.org/EllipticCurve/Q/40331/a/1 Mordell–Weil rank exactly 3 (Cremona's tables). Conductor 40331, minimal discriminant -40331, torsion trivial. The witnesses are the generators from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:36:15 UTC · history