curve #876

y² + xy + y = x³ + x² − 80x + 242

a-invariants: [1, 1, 1, -80, 242]
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: -0.4023
naive height: 24.7605
primes of bad reduction: 3, 5
regulator: 1
submitted by: David Renshaw
submitted at: 2026-09-23 14:41:29 UTC
last updated: 2026-09-23 14:41:29 UTC

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Commentary

LMFDB 15.a4 (Cremona 15a7): https://www.lmfdb.org/EllipticCurve/Q/15/a/4 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): #3 by |minimal discriminant|.

last edited by David Renshaw at 2026-09-23 14:41:29 UTC · history