curve #2453

y² + xy = x³ − 7132485696x + 93432847011840

a-invariants: [1, 0, 0, -7132485696, 93432847011840]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 21411390
discriminant (Δ): 19450887720043700961553947033600
Faltings height: 4.6973
naive height: 79.6774
primes of bad reduction: 2, 3, 5, 7, 11, 13, 23, 31
regulator: 3.057902184313398500758570638710523852582
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:02:31 UTC
last updated: 2026-09-26 20:02:31 UTC

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

(-88116, 6188196)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-31, q=8; A=(3q-p)(p+q)^3=-669185, B=-(p+3q)(p-q)^3=-415233. 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 20:02:31 UTC · history