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curve #656

y² + y = x³ − x² + 2

a-invariants: [0, -1, 1, 0, 2]
rank (lower bound): ≥ 2
torsion subgroup: trivial
conductor (N): 681
discriminant (Δ): -2043
Faltings height: -0.6857
naive height: 15.0781
primes of bad reduction: 3, 227
regulator: 0.136079688950694268446478372652388595164331
submitted by: David Renshaw
submitted at: 2026-09-08 19:03:56 UTC
last updated: 2026-09-08 19:03:56 UTC

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

(0, 1)
(3, 4)

Commentary

Cremona 681c1 = LMFDB 681.a1. Curve data, generators (0, 1) and (3, 4) and rank from the LMFDB: https://www.lmfdb.org/EllipticCurve/Q/681/a/1

last edited by David Renshaw at 2026-09-08 19:03:57 UTC · history

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