curve #2599

y² + xy = x³ − x² − 15106846114620x + 21144177172593648000

a-invariants: [1, -1, 0, -15106846114620, 21144177172593648000]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 1376845470
discriminant (Δ): 27511846929025637252307955040804432547600
Faltings height: 6.5063
naive height: 102.6521
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 29, 31
regulator: 13.5107263935341041449296755804427100446677
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:11:53 UTC
last updated: 2026-09-26 20:11:53 UTC

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

(1760148, 83032524)
(2725860, 466628880)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-60, q=31; A=(3q-p)(p+q)^3=-3731517, B=-(p+3q)(p-q)^3=24867843. 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 20:11:54 UTC · history