curve #1022

y² = x³ − x² − 6x

a-invariants: [0, -1, 0, -6, 0]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 480
discriminant (Δ): 14400 ☆ record for ℤ/2ℤ × ℤ/2ℤ torsion, rank ≥ 1
Faltings height: -0.5098 ☆ record for ℤ/2ℤ × ℤ/2ℤ torsion, rank ≥ 1
naive height: 17.1511 ☆ record for ℤ/2ℤ × ℤ/2ℤ torsion, rank ≥ 1
primes of bad reduction: 2, 3, 5
regulator: 1.035719522450406410025771591928353053720
submitted by: David Renshaw
submitted at: 2026-09-23 15:16:31 UTC
last updated: 2026-09-23 15:16:31 UTC

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

(6, 12)

Commentary

Cremona 480a1 = LMFDB 480.b3: https://www.lmfdb.org/EllipticCurve/Q/480/b/3 Mordell–Weil rank exactly 1 (Cremona's tables). Conductor 480, minimal discriminant 14400, torsion ℤ/2ℤ × ℤ/2ℤ. The witness is the generator from Cremona's tables.

last edited by David Renshaw at 2026-09-23 15:16:31 UTC · history