curve #2404

y² + xy + y = x³ − 60321830133x − 3181249158011444

a-invariants: [1, 0, 1, -60321830133, -3181249158011444]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 5704590
discriminant (Δ): 9675668556714471522220977539062500
Faltings height: 5.2195
naive height: 86.0825
primes of bad reduction: 2, 3, 5, 29, 79, 83
regulator: 4.509016688734142837181354477370980241157
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:59:31 UTC
last updated: 2026-09-26 19:59:31 UTC

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

(-174655, 45104292)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-2, q=27; A=(3q-p)(p+q)^3=1296875, B=-(p+3q)(p-q)^3=1926731. 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 19:59:31 UTC · history