curve #2664

y² + y = x³ − x² − 2x

a-invariants: [0, -1, 1, -2, 0]
rank (lower bound): ≥ 2
torsion subgroup: trivial
conductor (N): 709
discriminant (Δ): 709
Faltings height: -0.7604
naive height: 14.1555
primes of bad reduction: 709
regulator: 0.258683172736406669969244323486181605005230
submitted by: David Renshaw
submitted at: 2026-09-26 20:16:36 UTC
last updated: 2026-09-26 20:16:36 UTC

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

(0, 0)
(-1, 0)

Commentary

Cremona 709a1 = LMFDB 709.a1: https://www.lmfdb.org/EllipticCurve/Q/709/a/1 Mordell–Weil rank exactly 2 (Cremona's tables). Conductor 709, minimal discriminant 709, torsion trivial. The witnesses are the generators from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:16:36 UTC · history