curve #2408

y² + xy + y = x³ − 51570656638x + 3644177581212656

a-invariants: [1, 0, 1, -51570656638, 3644177581212656]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 23993970
discriminant (Δ): 3040859411561698709871352464000000
Faltings height: 5.1408
naive height: 85.6123
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 47
regulator: 8.643922919358779135052881607287070738256
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:59:46 UTC
last updated: 2026-09-26 19:59:46 UTC

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

(-1013285/4, 171113071/8)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-32, q=15; A=(3q-p)(p+q)^3=-378301, B=-(p+3q)(p-q)^3=1349699. 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 19:59:46 UTC · history