curve #869

y² + y = x³ + x² + x

a-invariants: [0, 1, 1, 1, 0]
rank (lower bound): ≥ 0
torsion subgroup: ℤ/3ℤ
conductor (N): 19 ☆ record for ℤ/3ℤ torsion, rank ≥ 0
discriminant (Δ): -19 ☆ record for ℤ/3ℤ torsion, rank ≥ 0
Faltings height: -1.0652 ☆ record for ℤ/3ℤ torsion, rank ≥ 0
naive height: 10.3972
primes of bad reduction: 19
regulator: 1
submitted by: David Renshaw
submitted at: 2026-09-23 14:40:24 UTC
last updated: 2026-09-23 14:40:24 UTC

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Commentary

LMFDB 19.a3 (Cremona 19a3): https://www.lmfdb.org/EllipticCurve/Q/19/a/3 Mordell–Weil rank exactly 0 per LMFDB. Conductor 19, minimal discriminant -19, torsion ℤ/3ℤ. 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): #4 by Faltings height, #8 by |minimal discriminant|.

last edited by David Renshaw at 2026-09-23 14:40:24 UTC · history