curve #2413

y² + xy + y = x³ − x² − 45234105152x + 1591565717650979

a-invariants: [1, -1, 1, -45234105152, 1591565717650979]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 5462730
discriminant (Δ): 4829218792354177360706793681000000
Faltings height: 5.1575
naive height: 85.2190
primes of bad reduction: 2, 3, 5, 7, 13, 23, 29
regulator: 4.889683961112287740367168180313598288464
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:00:04 UTC
last updated: 2026-09-26 20:00:04 UTC

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

(-228263, 4949881)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-39, q=10; A=(3q-p)(p+q)^3=-1682841, B=-(p+3q)(p-q)^3=-1058841. 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 20:00:04 UTC · history