curve #864

y² + xy + y = x³ − x

a-invariants: [1, 0, 1, -1, 0]
rank (lower bound): ≥ 0
torsion subgroup: ℤ/6ℤ
conductor (N): 14 ☆ record for ℤ/6ℤ torsion, rank ≥ 0
discriminant (Δ): -28 ☆ record for ℤ/6ℤ torsion, rank ≥ 0
Faltings height: -1.0321 ☆ record for ℤ/6ℤ torsion, rank ≥ 0
naive height: 11.0668 ☆ record for ℤ/6ℤ torsion, rank ≥ 0
primes of bad reduction: 2, 7
regulator: 1
submitted by: David Renshaw
submitted at: 2026-09-23 14:39:21 UTC
last updated: 2026-09-23 14:39:21 UTC

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Commentary

LMFDB 14.a5 (Cremona 14a4): https://www.lmfdb.org/EllipticCurve/Q/14/a/5 Mordell–Weil rank exactly 0 per LMFDB. Conductor 14, minimal discriminant -28, torsion ℤ/6ℤ. 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 conductor, #7 by Faltings height.

last edited by David Renshaw at 2026-09-23 14:39:21 UTC · history