curve #972

y² + xy + y = x³ + x² + x

a-invariants: [1, 1, 1, 1, 0]
rank (lower bound): ≥ 1
torsion subgroup: trivial
conductor (N): 83
discriminant (Δ): -83
Faltings height: -0.9439
naive height: 11.5504
primes of bad reduction: 83
regulator: 0.1772922941147684434576006159994158304290
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 83a1 = LMFDB 83.a1: https://www.lmfdb.org/EllipticCurve/Q/83/a/1 Mordell–Weil rank exactly 1 (Cremona's tables). Conductor 83, minimal discriminant -83, trivial torsion. The witness is the generator from Cremona's tables.

last edited by David Renshaw at 2026-09-23 15:08:58 UTC · history