curve #2697

y² + xy + y = x³ + x² − 544x + 4496

a-invariants: [1, 1, 1, -544, 4496]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 663
discriminant (Δ): 668584449
Faltings height: 0.4620
naive height: 30.5106
primes of bad reduction: 3, 13, 17
regulator: 2.975812071110832933108664309998461832386
submitted by: David Renshaw
submitted at: 2026-09-26 20:18:13 UTC
last updated: 2026-09-26 20:18:13 UTC

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

(79, 640)

Commentary

Cremona 663b2 = LMFDB 663.a3: https://www.lmfdb.org/EllipticCurve/Q/663/a/3 Mordell–Weil rank exactly 1 (Cremona's tables). Conductor 663, minimal discriminant 668584449, torsion ℤ/2ℤ × ℤ/4ℤ. The witness is the generator from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:18:13 UTC · history