curve #2597

y² + xy + y = x³ − x² − 56573161324817x + 163770570198642567809

a-invariants: [1, -1, 1, -56573161324817, 163770570198642567809]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 669278610
discriminant (Δ): 1492090506978762325517397463044096000000
Faltings height: 6.5665
naive height: 106.6133
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 23
regulator: 4.06870503472669441685528915501148229494623
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:11:46 UTC
last updated: 2026-09-26 20:11:46 UTC

Witness: 2 independent points log in to add more points →

(4298007, 121189876)
(4309767, 56321716)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-51, q=40; A=(3q-p)(p+q)^3=-227601, B=-(p+3q)(p-q)^3=51996399. The listed points are independent modulo torsion by the exact 2-descent Kummer squareclass map. The rank is a proven lower bound; no upper bound is claimed.

last edited by Natanael Wildner Fraga at 2026-09-26 20:11:46 UTC · history