curve #2313

y² + xy = x³ − x² − 1660253300064x + 822637016175321648

a-invariants: [1, -1, 0, -1660253300064, 822637016175321648]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 29047590
discriminant (Δ): 541196957238917365991005628906250000
Faltings height: 5.7729
naive height: 96.0276
primes of bad reduction: 2, 3, 5, 11, 13, 37, 61
regulator: 6.923898090176794558248721544930889927630
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:53:48 UTC
last updated: 2026-09-26 19:53:48 UTC

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

(809187, 94587594)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-36, q=25; A=(3q-p)(p+q)^3=-147741, B=-(p+3q)(p-q)^3=8852259. 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 19:53:48 UTC · history