curve #2747

y² + xy + y = x³ + x² − 3x − 2

a-invariants: [1, 1, 1, -3, -2]
rank (lower bound): ≥ 2
torsion subgroup: trivial
conductor (N): 1627
discriminant (Δ): 1627
Faltings height: -0.6925
naive height: 14.9302
primes of bad reduction: 1627
regulator: 0.479877399706461244321122652383092454328435
submitted by: David Renshaw
submitted at: 2026-09-26 20:20:29 UTC
last updated: 2026-09-26 20:20:29 UTC

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

(-1, 1)
(-2, 1)

Commentary

Cremona 1627a1 = LMFDB 1627.a1: https://www.lmfdb.org/EllipticCurve/Q/1627/a/1 Mordell–Weil rank exactly 2 (Cremona's tables). Conductor 1627, minimal discriminant 1627, torsion trivial. The witnesses are the generators from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:20:29 UTC · history