curve #2259

y² + xy + y = x³ − 8832257555908x + 9469663691710898306

a-invariants: [1, 0, 1, -8832257555908, 9469663691710898306]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 458948490
discriminant (Δ): 5356185916683904925295713659539921000000
Faltings height: 6.3707
naive height: 101.0419
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 29, 31
regulator: 5.069501224186913668642599635897827935743
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:50:20 UTC
last updated: 2026-09-26 19:50:20 UTC

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

(1305460, 404691317)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-56, q=29; A=(3q-p)(p+q)^3=-2814669, B=-(p+3q)(p-q)^3=19037875. 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 19:50:20 UTC · history