curve #867

y² + xy + y = x³ + x²

a-invariants: [1, 1, 1, 0, 0]
rank (lower bound): ≥ 0
torsion subgroup: ℤ/4ℤ
conductor (N): 15 ☆ record for ℤ/4ℤ torsion, rank ≥ 0
discriminant (Δ): -15 ☆ record for ℤ/4ℤ torsion, rank ≥ 0
Faltings height: -1.0954 ☆ record for ℤ/4ℤ torsion, rank ≥ 0
naive height: 10.1628 ☆ record for ℤ/4ℤ torsion, rank ≥ 0
primes of bad reduction: 3, 5
regulator: 1
submitted by: David Renshaw
submitted at: 2026-09-23 14:40:23 UTC
last updated: 2026-09-23 14:40:23 UTC

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Commentary

LMFDB 15.a7 (Cremona 15a8): https://www.lmfdb.org/EllipticCurve/Q/15/a/7 Mordell–Weil rank exactly 0 per LMFDB. Conductor 15, minimal discriminant -15, torsion ℤ/4ℤ. 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): #2 by Faltings height, #4 by |minimal discriminant|.

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