curve #887

y² + y = x³ + x² − 9x − 15

a-invariants: [0, 1, 1, -9, -15]
rank (lower bound): ≥ 0
torsion subgroup: ℤ/3ℤ
conductor (N): 19 ☆ record for ℤ/3ℤ torsion, rank ≥ 0
discriminant (Δ): -6859
Faltings height: -0.5159
naive height: 18.4382
primes of bad reduction: 19
regulator: 1
submitted by: David Renshaw
submitted at: 2026-09-23 14:56:04 UTC
last updated: 2026-09-23 14:56:04 UTC

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Commentary

Cremona 19a1 = LMFDB 19.a2: https://www.lmfdb.org/EllipticCurve/Q/19/a/2 Mordell–Weil rank exactly 0 (Cremona's tables). Conductor 19, minimal discriminant -6859, torsion ℤ/3ℤ.

last edited by David Renshaw at 2026-09-23 14:56:04 UTC · history