curve #2349

y² + xy + y = x³ − 288251349123x − 40935414215505494

a-invariants: [1, 0, 1, -288251349123, -40935414215505494]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 701563170
discriminant (Δ): 808925033923658799857567978490562500
Faltings height: 5.5949
naive height: 90.7749
primes of bad reduction: 2, 3, 5, 7, 11, 31, 97, 101
regulator: 5.886110154197880447945702426024071355512
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:56:03 UTC
last updated: 2026-09-26 19:56:03 UTC

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

(-384655, 114335202)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-2, q=33; A=(3q-p)(p+q)^3=3008891, B=-(p+3q)(p-q)^3=4158875. 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 19:56:03 UTC · history