curve #3289

y² + xy + y = x³ + x² − 539x + 4592

a-invariants: [1, 1, 1, -539, 4592]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/4ℤ
conductor (N): 663
discriminant (Δ): 25857
Faltings height: 0.1154
naive height: 30.4829
primes of bad reduction: 3, 13, 17
regulator: 1.487906035555416466554332154999230916193
submitted by: David Renshaw
submitted at: 2026-09-26 20:52:15 UTC
last updated: 2026-09-26 20:52:15 UTC

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

(9, 19)

Commentary

Cremona 663b1 = LMFDB 663.a4: https://www.lmfdb.org/EllipticCurve/Q/663/a/4 Mordell–Weil rank exactly 1 (Cremona's tables). Conductor 663, minimal discriminant 25857, torsion ℤ/4ℤ. The witness is the generator from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:52:15 UTC · history