curve #875

y² = x³ + x² + x

a-invariants: [0, 1, 0, 1, 0]
rank (lower bound): ≥ 0
torsion subgroup: ℤ/2ℤ
conductor (N): 48
discriminant (Δ): -48
Faltings height: -0.9919
naive height: 10.8233 ☆ record for ℤ/2ℤ torsion, rank ≥ 0
primes of bad reduction: 2, 3
regulator: 1
submitted by: David Renshaw
submitted at: 2026-09-23 14:41:28 UTC
last updated: 2026-09-23 14:41:28 UTC

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Commentary

LMFDB 48.a5 (Cremona 48a4): https://www.lmfdb.org/EllipticCurve/Q/48/a/5 Mordell–Weil rank exactly 0 per LMFDB. Conductor 48, minimal discriminant -48, torsion ℤ/2ℤ. Submitted as one of the ten smallest rank-0 curves in LMFDB (as of 2026-09-23; places in LMFDB sort order, ties broken by label): #10 by Faltings height.

last edited by David Renshaw at 2026-09-23 14:41:28 UTC · history