curve #2511

y² + xy = x³ − x² − 85049424x + 297674459568

a-invariants: [1, -1, 0, -85049424, 297674459568]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 131670
discriminant (Δ): 1098513156172746872250000
Faltings height: 3.4126
naive height: 66.3898
primes of bad reduction: 2, 3, 5, 7, 11, 19
regulator: 2.521151845915205589387793252125812550149
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:06:08 UTC
last updated: 2026-09-26 20:06:08 UTC

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

(-8868, 599784)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-12, q=7; A=(3q-p)(p+q)^3=-4125, B=-(p+3q)(p-q)^3=61731. 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 20:06:09 UTC · history