curve #3333

y² = x³ − 4x + 4

a-invariants: [0, 0, 0, -4, 4]
rank (lower bound): ≥ 1
torsion subgroup: trivial
conductor (N): 88
discriminant (Δ): -2816
Faltings height: -0.6292
naive height: 16.2957
primes of bad reduction: 2, 11
regulator: 0.04026436433688064294321515224808125270452
submitted by: David Renshaw
submitted at: 2026-09-26 20:59:21 UTC
last updated: 2026-09-26 20:59:21 UTC

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

(2, 2)

Commentary

Cremona 88a1 = LMFDB 88.a1: https://www.lmfdb.org/EllipticCurve/Q/88/a/1 Mordell–Weil rank exactly 1 (Cremona's tables). Conductor 88, minimal discriminant -2816, torsion trivial. The witness is the generator from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:59:21 UTC · history