curve #3076

y² + y = x³ − x² − 5x − 3

a-invariants: [0, -1, 1, -5, -3]
rank (lower bound): ≥ 2
torsion subgroup: trivial
conductor (N): 997
discriminant (Δ): 997
Faltings height: -0.6716
naive height: 16.6355
primes of bad reduction: 997
regulator: 0.571019259287366988341108843169484267450998
submitted by: David Renshaw
submitted at: 2026-09-26 20:36:06 UTC
last updated: 2026-09-26 20:36:06 UTC

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

(-1, 0)
(-3/4, -3/8)

Commentary

Cremona 997b1 = LMFDB 997.c1: https://www.lmfdb.org/EllipticCurve/Q/997/c/1 Mordell–Weil rank exactly 2 (Cremona's tables). Conductor 997, minimal discriminant 997, torsion trivial. The witnesses are the generators from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:36:06 UTC · history