curve #995

y² + xy + y = x³ + x² + 3x + 3

a-invariants: [1, 1, 1, 3, 3]
rank (lower bound): ≥ 0
torsion subgroup: ℤ/4ℤ
conductor (N): 138
discriminant (Δ): -3312
Faltings height: -0.6381
naive height: 14.8885
primes of bad reduction: 2, 3, 23
regulator: 1
submitted by: David Renshaw
submitted at: 2026-09-23 15:12:12 UTC
last updated: 2026-09-23 15:12:12 UTC

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Commentary

Cremona 138c1 = LMFDB 138.c4: https://www.lmfdb.org/EllipticCurve/Q/138/c/4 Mordell–Weil rank exactly 0 (Cremona's tables). Conductor 138, minimal discriminant -3312, torsion ℤ/4ℤ.

last edited by David Renshaw at 2026-09-23 15:12:13 UTC · history