curve #2324

y² + xy = x³ − 589578568156x − 53232308371671280

a-invariants: [1, 0, 0, -589578568156, -53232308371671280]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 219609390
discriminant (Δ): 11891962713617683242354672744687206400
Faltings height: 5.8058
naive height: 92.9216
primes of bad reduction: 2, 3, 5, 7, 11, 13, 71, 103
regulator: 4.901824959710835021042315490261775263884
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:54:27 UTC
last updated: 2026-09-26 19:54:27 UTC

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

(811244, 48267860)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-55, q=16; A=(3q-p)(p+q)^3=-6109857, B=-(p+3q)(p-q)^3=-2505377. 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 19:54:27 UTC · history