curve #2333

y² + xy = x³ − 572983039145x + 166352802519632025

a-invariants: [1, 0, 0, -572983039145, 166352802519632025]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 206308410
discriminant (Δ): 84558939161924101152830770449000000
Faltings height: 5.5579
naive height: 92.8360
primes of bad reduction: 2, 3, 5, 7, 11, 31, 43, 67
regulator: 6.327653640866345186328553439485940092016
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:55:02 UTC
last updated: 2026-09-26 19:55:02 UTC

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

(-2669645/4, 4014476675/8)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-49, q=6; A=(3q-p)(p+q)^3=-5326969, B=-(p+3q)(p-q)^3=-5157625. 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 19:55:03 UTC · history