curve #1012

y² + y = x³ + x² + 3x + 2

a-invariants: [0, 1, 1, 3, 2]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/3ℤ
conductor (N): 219
discriminant (Δ): -1971
Faltings height: -0.6806
naive height: 14.5561 ☆ record for ℤ/3ℤ torsion, rank ≥ 1
primes of bad reduction: 3, 73
regulator: 1.256236863853751532911588770361478097847
submitted by: David Renshaw
submitted at: 2026-09-23 15:14:23 UTC
last updated: 2026-09-23 15:14:23 UTC

Witness: 1 independent point log in to add more points →

(2, 4)

Commentary

Cremona 219b1 = LMFDB 219.b2: https://www.lmfdb.org/EllipticCurve/Q/219/b/2 Mordell–Weil rank exactly 1 (Cremona's tables). Conductor 219, minimal discriminant -1971, torsion ℤ/3ℤ. The witness is the generator from Cremona's tables.

last edited by David Renshaw at 2026-09-23 15:14:24 UTC · history