curve #2399

y² + xy = x³ − x² − 65952825939x − 4830952408589727

a-invariants: [1, -1, 0, -65952825939, -4830952408589727]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 2612610
discriminant (Δ): 8278189545197209746755782197562500
Faltings height: 5.2172
naive height: 86.3502
primes of bad reduction: 2, 3, 5, 7, 11, 13, 29
regulator: 7.503463014608901282331618912790419339742
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:59:10 UTC
last updated: 2026-09-26 19:59:10 UTC

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

(-90348, 19800129)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-42, q=13; A=(3q-p)(p+q)^3=-1975509, B=-(p+3q)(p-q)^3=-499125. 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 19:59:11 UTC · history