curve #3065

y² + xy + y = x³ + x² + 3x

a-invariants: [1, 1, 1, 3, 0]
rank (lower bound): ≥ 2
torsion subgroup: trivial
conductor (N): 2181
discriminant (Δ): -2181
Faltings height: -0.6712
naive height: 14.8885
primes of bad reduction: 3, 727
regulator: 0.745548011675010499223838574638368248393567
submitted by: David Renshaw
submitted at: 2026-09-26 20:35:37 UTC
last updated: 2026-09-26 20:35:37 UTC

Witness: 2 independent points log in to add more points →

(0, 0)
(2, 3)

Commentary

Cremona 2181a1 = LMFDB 2181.a1: https://www.lmfdb.org/EllipticCurve/Q/2181/a/1 Mordell–Weil rank exactly 2 (Cremona's tables). Conductor 2181, minimal discriminant -2181, torsion trivial. The witnesses are the generators from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:35:37 UTC · history