curve #939

y² = x³ + x² − 4x − 4

a-invariants: [0, 1, 0, -4, -4]
rank (lower bound): ≥ 0
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 48
discriminant (Δ): 2304
Faltings height: -0.6454
naive height: 16.0126
primes of bad reduction: 2, 3
regulator: 1
submitted by: David Renshaw
submitted at: 2026-09-23 15:03:36 UTC
last updated: 2026-09-23 15:03:36 UTC

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Commentary

Cremona 48a1 = LMFDB 48.a4: https://www.lmfdb.org/EllipticCurve/Q/48/a/4 Mordell–Weil rank exactly 0 (Cremona's tables). Conductor 48, minimal discriminant 2304, torsion ℤ/2ℤ × ℤ/2ℤ.

last edited by David Renshaw at 2026-09-23 15:03:36 UTC · history