curve #642

y² + y = x³ − x² − 2x + 2

a-invariants: [0, -1, 1, -2, 2]
rank (lower bound): ≥ 1
torsion subgroup: trivial
conductor (N): 57
discriminant (Δ): -171
Faltings height: -0.8365
naive height: 14.3464
primes of bad reduction: 3, 19
regulator: 0.03757459273682375282480511121671040840484
submitted by: David Renshaw
submitted at: 2026-09-08 17:17:39 UTC
last updated: 2026-09-08 17:17:39 UTC

Witness: 1 independent point log in to add more points →

(2, 1)

Commentary

Cremona 57a1 = LMFDB 57.a1, the rank-1 curve of smallest conductor not previously on the board. Curve data, generator (2, 1) and rank from the LMFDB: https://www.lmfdb.org/EllipticCurve/Q/57/a/1

last edited by David Renshaw at 2026-09-08 17:17:39 UTC · history