curve #2640

y² + xy + y = x³ − 2007239907718x + 1094578616238540056

a-invariants: [1, 0, 1, -2007239907718, 1094578616238540056]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 137244030
discriminant (Δ): 29110997535551241947498169000000
Faltings height: 5.5387
naive height: 96.5969
primes of bad reduction: 2, 3, 5, 7, 11, 19, 53, 59
regulator: 26.7342935825022993019736722373260034289017
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:14:38 UTC
last updated: 2026-09-26 20:14:38 UTC

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

(-5410985/4, 9246864181/8)
(802597, 23608973)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-56, q=1; A=(3q-p)(p+q)^3=-9816125, B=-(p+3q)(p-q)^3=-9815229. 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 20:14:38 UTC · history