curve #2351

y² + xy + y = x³ − 268144194193x + 52125335330691056

a-invariants: [1, 0, 1, -268144194193, 52125335330691056]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 1074902010
discriminant (Δ): 60147927617060698492948533680062500
Faltings height: 5.4523
naive height: 90.5580
primes of bad reduction: 2, 3, 5, 7, 13, 17, 19, 23, 53
regulator: 5.418240140522622952063117390157930661825
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:56:10 UTC
last updated: 2026-09-26 19:56:10 UTC

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

(-593480, 47509007)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-34, q=19; A=(3q-p)(p+q)^3=-307125, B=-(p+3q)(p-q)^3=3424171. 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 19:56:10 UTC · history