curve #3068

y² + xy = x³ + x − 2

a-invariants: [1, 0, 0, 1, -2]
rank (lower bound): ≥ 2
torsion subgroup: trivial
conductor (N): 1933
discriminant (Δ): -1933
Faltings height: -0.6890
naive height: 14.9900
primes of bad reduction: 1933
regulator: 0.755018996646126740980464274379304186894051
submitted by: David Renshaw
submitted at: 2026-09-26 20:35:46 UTC
last updated: 2026-09-26 20:35:46 UTC

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

(1, 0)
(2, 2)

Commentary

Cremona 1933a1 = LMFDB 1933.a1: https://www.lmfdb.org/EllipticCurve/Q/1933/a/1 Mordell–Weil rank exactly 2 (Cremona's tables). Conductor 1933, minimal discriminant -1933, torsion trivial. The witnesses are the generators from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:35:46 UTC · history