curve #974

y² + xy + y = x³ + x² − x

a-invariants: [1, 1, 1, -1, 0]
rank (lower bound): ≥ 1
torsion subgroup: trivial
conductor (N): 89
discriminant (Δ): -89
Faltings height: -0.9269
naive height: 12.5115
primes of bad reduction: 89
regulator: 0.1121048812300989494200897750437008367331
submitted by: David Renshaw
submitted at: 2026-09-23 15:08:58 UTC
last updated: 2026-09-23 15:08:58 UTC

Witness: 1 independent point log in to add more points →

(0, 0)

Commentary

Cremona 89a1 = LMFDB 89.a1: https://www.lmfdb.org/EllipticCurve/Q/89/a/1 Mordell–Weil rank exactly 1 (Cremona's tables). Conductor 89, minimal discriminant -89, trivial torsion. The witness is the generator from Cremona's tables.

last edited by David Renshaw at 2026-09-23 15:08:59 UTC · history