curve #2362

y² + xy = x³ − x² − 196074113499x + 18743066313400593

a-invariants: [1, -1, 0, -196074113499, 18743066313400593]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 13026510
discriminant (Δ): 330675261174527779289636788392562500
Faltings height: 5.5139
naive height: 89.6189
primes of bad reduction: 2, 3, 5, 7, 23, 29, 31
regulator: 5.306914798037420272856372045031311186720
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:56:51 UTC
last updated: 2026-09-26 19:56:51 UTC

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

(-477471, 59479341)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-6, q=29; A=(3q-p)(p+q)^3=1131531, B=-(p+3q)(p-q)^3=3472875. 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 19:56:51 UTC · history