curve #2224

y² + xy = x³ − 49139393567145x + 132522794312863952025

a-invariants: [1, 0, 0, -49139393567145, 132522794312863952025]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 2291018730
discriminant (Δ): 7068109153274087217112854466186761000000
Faltings height: 6.5955
naive height: 106.1906
primes of bad reduction: 2, 3, 5, 7, 11, 13, 23, 31, 107
regulator: 2.250476879263995517758040599242136267675
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:48:10 UTC
last updated: 2026-09-26 19:48:10 UTC

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

(3964926, 139106667)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-31, q=46; A=(3q-p)(p+q)^3=570375, B=-(p+3q)(p-q)^3=48849031. 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 19:48:10 UTC · history