curve #2407

y² + xy = x³ − 51906895251x + 4451209707617505

a-invariants: [1, 0, 0, -51906895251, 4451209707617505]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 309242670
discriminant (Δ): 391313136744200032758778981262400
Faltings height: 5.0367
naive height: 85.6318
primes of bad reduction: 2, 3, 5, 11, 19, 31, 37, 43
regulator: 6.217396926375042195305449555274026591660
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:59:42 UTC
last updated: 2026-09-26 19:59:42 UTC

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

(-252420, 38472105)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-37, q=6; A=(3q-p)(p+q)^3=-1638505, B=-(p+3q)(p-q)^3=-1510633. 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 19:59:42 UTC · history