curve #963

y² + xy = x³ − 45x + 81

a-invariants: [1, 0, 0, -45, 81]
rank (lower bound): ≥ 0
torsion subgroup: ℤ/10ℤ
conductor (N): 66 ☆ record for ℤ/10ℤ torsion, rank ≥ 0
discriminant (Δ): 2737152 ☆ record for ℤ/10ℤ torsion, rank ≥ 0
Faltings height: -0.0569 ☆ record for ℤ/10ℤ torsion, rank ≥ 0
naive height: 23.0350 ☆ record for ℤ/10ℤ torsion, rank ≥ 0
primes of bad reduction: 2, 3, 11
regulator: 1
submitted by: David Renshaw
submitted at: 2026-09-23 15:06:50 UTC
last updated: 2026-09-23 15:06:50 UTC

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Commentary

Cremona 66c1 = LMFDB 66.c3: https://www.lmfdb.org/EllipticCurve/Q/66/c/3 Mordell–Weil rank exactly 0 (Cremona's tables). Conductor 66, minimal discriminant 2737152, torsion ℤ/10ℤ.

last edited by David Renshaw at 2026-09-23 15:06:51 UTC · history