curve #2325

y² + xy = x³ − x² − 2736009201024x + 1741833502589400768

a-invariants: [1, -1, 0, -2736009201024, 1741833502589400768]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 64234170
discriminant (Δ): 108499298044104642740159343249000000
Faltings height: 5.7959
naive height: 97.5262
primes of bad reduction: 2, 3, 5, 7, 11, 13, 23, 31
regulator: 3.017110715002690897593826278415168216080
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:54:30 UTC
last updated: 2026-09-26 19:54:30 UTC

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

(941232, 21150984)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-24, q=31; A=(3q-p)(p+q)^3=40131, B=-(p+3q)(p-q)^3=11479875. 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 19:54:31 UTC · history