curve #2209

y² + xy = x³ − 56732328498495x + 152281817756298056025

a-invariants: [1, 0, 0, -56732328498495, 152281817756298056025]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 7538941410
discriminant (Δ): 1668185770350604092654789636225152841000000
Faltings height: 6.8440
naive height: 106.6217
primes of bad reduction: 2, 3, 5, 7, 11, 13, 19, 73, 181
regulator: 3.469427876975316033505372885565812678417
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:47:14 UTC
last updated: 2026-09-26 19:47:14 UTC

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

(5508960, 2630711355)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-19, q=54; A=(3q-p)(p+q)^3=7760375, B=-(p+3q)(p-q)^3=55629431. 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 19:47:15 UTC · history