curve #3256

y² = x³ + x² − 157x + 716

a-invariants: [0, 1, 0, -157, 716]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/3ℤ
conductor (N): 5988
discriminant (Δ): -5820336
Faltings height: 0.1053
naive height: 26.8118
primes of bad reduction: 2, 3, 499
regulator: 0.979996911166885960715488978920846292538714
submitted by: David Renshaw
submitted at: 2026-09-26 20:46:52 UTC
last updated: 2026-09-26 20:46:52 UTC

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

(8, 6)
(7, 3)

Commentary

Cremona 5988b1 = LMFDB 5988.a1: https://www.lmfdb.org/EllipticCurve/Q/5988/a/1 Mordell–Weil rank exactly 2 (Cremona's tables). Conductor 5988, minimal discriminant -5820336, torsion ℤ/3ℤ. The witnesses are the generators from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:46:52 UTC · history