curve #1054

y² + xy = x³ − 31900x + 1610000

a-invariants: [1, 0, 0, -31900, 1610000]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 4290
discriminant (Δ): 954068544000000 ☆ record for ℤ/2ℤ × ℤ/6ℤ torsion, rank ≥ 1
Faltings height: 1.5828 ☆ record for ℤ/2ℤ × ℤ/6ℤ torsion, rank ≥ 1
naive height: 42.7247 ☆ record for ℤ/2ℤ × ℤ/6ℤ torsion, rank ≥ 1
primes of bad reduction: 2, 3, 5, 11, 13
regulator: 1.072602621112355346232571397275582832293
submitted by: David Renshaw
submitted at: 2026-09-23 15:21:51 UTC
last updated: 2026-09-23 15:21:51 UTC

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

(50, 350)

Commentary

Cremona 4290bb2 = LMFDB 4290.bb6: https://www.lmfdb.org/EllipticCurve/Q/4290/bb/6 Mordell–Weil rank exactly 1 (Cremona's tables). Conductor 4290, minimal discriminant 954068544000000, torsion ℤ/2ℤ × ℤ/6ℤ. The witness is the generator from Cremona's tables.

last edited by David Renshaw at 2026-09-23 15:21:51 UTC · history