curve #2245

y² + xy = x³ − x² − 22360369248414x + 40682937371450979348

a-invariants: [1, -1, 0, -22360369248414, 40682937371450979348]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 219565710
discriminant (Δ): 508337304751811587722222425021604000000
Faltings height: 6.3894
naive height: 103.8285
primes of bad reduction: 2, 3, 5, 7, 13, 17, 19, 83
regulator: 8.458088842867267101995887799847010035909
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:49:28 UTC
last updated: 2026-09-26 19:49:28 UTC

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

(11836551/4, 5216985663/8)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-48, q=35; A=(3q-p)(p+q)^3=-336141, B=-(p+3q)(p-q)^3=32591859. 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 19:49:28 UTC · history