curve #2507

y² + xy = x³ − 116289615x − 412420328775

a-invariants: [1, 0, 0, -116289615, -412420328775]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 274170
discriminant (Δ): 27171794302813265625000000
Faltings height: 3.6034
naive height: 67.3284
primes of bad reduction: 2, 3, 5, 13, 19, 37
regulator: 4.658237581042813883072855951829358631663
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:05:53 UTC
last updated: 2026-09-26 20:05:53 UTC

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

(-22305/4, 2024805/8)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-19, q=6; A=(3q-p)(p+q)^3=-81289, B=-(p+3q)(p-q)^3=-15625. 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 20:05:54 UTC · history