curve #2631

y² + xy = x³ − 163796520041x + 25134105619020825

a-invariants: [1, 0, 0, -163796520041, 25134105619020825]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 33090330
discriminant (Δ): 8346195162194017939784779335566400
Faltings height: 5.3062
naive height: 89.0793
primes of bad reduction: 2, 3, 5, 7, 13, 17, 23, 31
regulator: 9.77138801053212113687862165050606914165457
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:13:59 UTC
last updated: 2026-09-26 20:13:59 UTC

Witness: 2 independent points log in to add more points →

(-464492, 31868851)
(-396242, 167003851)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-31, q=18; A=(3q-p)(p+q)^3=-186745, B=-(p+3q)(p-q)^3=2705927. 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 20:13:59 UTC · history