curve #915

y² = x³ − x

a-invariants: [0, 0, 0, -1, 0]
rank (lower bound): ≥ 0
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 32
discriminant (Δ): 64 ☆ record for ℤ/2ℤ × ℤ/2ℤ torsion, rank ≥ 0
Faltings height: -0.9640 ☆ record for ℤ/2ℤ × ℤ/2ℤ torsion, rank ≥ 0
naive height: 11.6136 ☆ record for ℤ/2ℤ × ℤ/2ℤ torsion, rank ≥ 0
primes of bad reduction: 2
regulator: 1
submitted by: David Renshaw
submitted at: 2026-09-23 15:00:22 UTC
last updated: 2026-09-23 15:00:22 UTC

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Commentary

Cremona 32a2 = LMFDB 32.a3: https://www.lmfdb.org/EllipticCurve/Q/32/a/3 Mordell–Weil rank exactly 0 (Cremona's tables). Conductor 32, minimal discriminant 64, torsion ℤ/2ℤ × ℤ/2ℤ.

last edited by David Renshaw at 2026-09-23 15:00:22 UTC · history