curve #2406

y² + xy = x³ − 485338725185x + 130140694213521225

a-invariants: [1, 0, 0, -485338725185, 130140694213521225]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 21447510
discriminant (Δ): 76726732796236832464230201000000
Faltings height: 5.3058
naive height: 92.3379
primes of bad reduction: 2, 3, 5, 7, 41, 47, 53
regulator: 8.581508558985585666121570426126286076756
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:59:38 UTC
last updated: 2026-09-26 19:59:38 UTC

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

(-703988, 350948107)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-47, q=2; A=(3q-p)(p+q)^3=-4829625, B=-(p+3q)(p-q)^3=-4823609. 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 19:59:38 UTC · history