curve #2193

y² + xy = x³ − 131017936706945x + 575516124248439854025

a-invariants: [1, 0, 0, -131017936706945, 575516124248439854025]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 9193657290
discriminant (Δ): 850400929733720458422857990186618721000000
Faltings height: 6.9090
naive height: 109.1327
primes of bad reduction: 2, 3, 5, 17, 23, 31, 131, 193
regulator: 4.403498015035708341992324406165870989522
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:46:13 UTC
last updated: 2026-09-26 19:46:13 UTC

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

(6299992, 384443221)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-31, q=54; A=(3q-p)(p+q)^3=2348231, B=-(p+3q)(p-q)^3=80450375. 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 19:46:13 UTC · history