curve #3185

y² + xy + y = x³ − x² + x − 2

a-invariants: [1, -1, 1, 1, -2]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ
conductor (N): 333
discriminant (Δ): -999
Faltings height: -0.7417
naive height: 14.2050
primes of bad reduction: 3, 37
regulator: 0.9605876837473819112921810576819444984411
submitted by: David Renshaw
submitted at: 2026-09-26 20:41:37 UTC
last updated: 2026-09-26 20:41:37 UTC

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

(2, 1)

Commentary

Cremona 333c1 = LMFDB 333.a2: https://www.lmfdb.org/EllipticCurve/Q/333/a/2 Mordell–Weil rank exactly 1 (Cremona's tables). Conductor 333, minimal discriminant -999, torsion ℤ/2ℤ. The witness is the generator from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:41:38 UTC · history