curve #2506

y² + xy = x³ − 263685125x + 1647612890625

a-invariants: [1, 0, 0, -263685125, 1647612890625]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 3038070
discriminant (Δ): 624419211866828049000000
Faltings height: 3.5426
naive height: 69.7844
primes of bad reduction: 2, 3, 5, 7, 17, 23, 37
regulator: 4.552000068653776600231922546416867224788
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:05:50 UTC
last updated: 2026-09-26 20:05:50 UTC

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

(12975/4, 7259625/8)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-7, q=10; A=(3q-p)(p+q)^3=999, B=-(p+3q)(p-q)^3=112999. 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 20:05:50 UTC · history