curve #2609

y² + xy = x³ − 3267735705375x + 2082498719788265625

a-invariants: [1, 0, 0, -3267735705375, 2082498719788265625]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 1615817190
discriminant (Δ): 359666163515140211088355838765625000000
Faltings height: 6.1367
naive height: 98.0590
primes of bad reduction: 2, 3, 5, 13, 17, 19, 101, 127
regulator: 26.2314714362678951965304923824865213742089
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:12:31 UTC
last updated: 2026-09-26 20:12:31 UTC

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

(-1963968, 962683077)
(694056, 385455477)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-13, q=38; A=(3q-p)(p+q)^3=1984375, B=-(p+3q)(p-q)^3=13397751. 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 20:12:31 UTC · history