curve #872

y² + xy = x³ + x² + x

a-invariants: [1, 1, 0, 1, 0]
rank (lower bound): ≥ 0
torsion subgroup: ℤ/2ℤ
conductor (N): 39
discriminant (Δ): -39
Faltings height: -1.0115 ☆ record for ℤ/2ℤ torsion, rank ≥ 0
naive height: 10.9192
primes of bad reduction: 3, 13
regulator: 1
submitted by: David Renshaw
submitted at: 2026-09-23 14:40:26 UTC
last updated: 2026-09-23 14:40:26 UTC

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Commentary

LMFDB 39.a4 (Cremona 39a4): https://www.lmfdb.org/EllipticCurve/Q/39/a/4 Mordell–Weil rank exactly 0 per LMFDB. Conductor 39, minimal discriminant -39, 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): #8 by Faltings height.

last edited by David Renshaw at 2026-09-23 14:40:26 UTC · history