curve #952

y² + xy + y = x³ − 2x − 1

a-invariants: [1, 0, 1, -2, -1]
rank (lower bound): ≥ 0
torsion subgroup: ℤ/2ℤ
conductor (N): 57
discriminant (Δ): 57
Faltings height: -0.9382
naive height: 12.8714
primes of bad reduction: 3, 19
regulator: 1
submitted by: David Renshaw
submitted at: 2026-09-23 15:05:44 UTC
last updated: 2026-09-23 15:05:44 UTC

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Commentary

Cremona 57b2 = LMFDB 57.c3: https://www.lmfdb.org/EllipticCurve/Q/57/c/3 Mordell–Weil rank exactly 0 (Cremona's tables). Conductor 57, minimal discriminant 57, torsion ℤ/2ℤ.

last edited by David Renshaw at 2026-09-23 15:05:44 UTC · history