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curve #653

y² + xy = x³ − 4x + 3

a-invariants: [1, 0, 0, -4, 3]
rank (lower bound): ≥ 2
torsion subgroup: trivial
conductor (N): 643
discriminant (Δ): -643
Faltings height: -0.7177
naive height: 15.9318
primes of bad reduction: 643
regulator: 0.229179621568847078938767135198384435021017
submitted by: David Renshaw
submitted at: 2026-09-08 19:03:43 UTC
last updated: 2026-09-08 19:03:43 UTC

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

(1, 0)
(2, 1)

Commentary

Cremona 643a1 = LMFDB 643.a1. Curve data, generators (1, 0) and (2, 1) and rank from the LMFDB: https://www.lmfdb.org/EllipticCurve/Q/643/a/1

last edited by David Renshaw at 2026-09-08 19:03:43 UTC · history

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