curve #882

y² + xy + y = x³ + x² − 5x + 2

a-invariants: [1, 1, 1, -5, 2]
rank (lower bound): ≥ 0
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 15 ☆ record for ℤ/2ℤ × ℤ/4ℤ torsion, rank ≥ 0
discriminant (Δ): 225 ☆ record for ℤ/2ℤ × ℤ/4ℤ torsion, rank ≥ 0
Faltings height: -0.7489 ☆ record for ℤ/2ℤ × ℤ/4ℤ torsion, rank ≥ 0
naive height: 16.4544
primes of bad reduction: 3, 5
regulator: 1
submitted by: David Renshaw
submitted at: 2026-09-23 14:55:00 UTC
last updated: 2026-09-23 14:55:00 UTC

Witness: no points (rank ≥ 0 needs none) log in to add more points →

Commentary

Cremona 15a3 = LMFDB 15.a6: https://www.lmfdb.org/EllipticCurve/Q/15/a/6 Mordell–Weil rank exactly 0 (Cremona's tables). Conductor 15, minimal discriminant 225, torsion ℤ/2ℤ × ℤ/4ℤ.

last edited by David Renshaw at 2026-09-23 14:55:00 UTC · history