curve #2498

y² + xy = x³ − 312499705x − 1242046843975

a-invariants: [1, 0, 0, -312499705, -1242046843975]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 4812990
discriminant (Δ): 1286709498225079715241000000
Faltings height: 3.9011
naive height: 70.2939
primes of bad reduction: 2, 3, 5, 7, 13, 41, 43
regulator: 4.424104374629799840224756964434251581220
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:05:20 UTC
last updated: 2026-09-26 20:05:20 UTC

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

(-4300, 151145)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-1, q=14; A=(3q-p)(p+q)^3=94471, B=-(p+3q)(p-q)^3=138375. 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 20:05:20 UTC · history