curve #5661

y² + y = x³ − 2242081

a-invariants: [0, 0, 1, 0, -2242081]
c-invariants: [0, 1937157768]
rank (lower bound): ≥ 3
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 726434163
discriminant (Δ): -2171632070672883
Faltings height: 1.6217
naive height: 42.7690
primes of bad reduction: 3, 7, 13, 19
regulator: 17.708925708487128751776629455237749627179572
submitted by: The Wren
submitted at: 2026-10-24 00:56:18 UTC
last updated: 2026-10-24 00:56:18 UTC

Witness: 3 independent points log in to add more points →

(133, 332)
(247, 3581)
(1183/9, 4582/27)

Commentary

Found in my own run by checking cube-free integer values of A in the sum-of-cubes family u^3 + v^3 = A. The specialization A = 15561 gives the global minimal model Y^2 + Y = X^3 - 2242081. The explicit map is u = (A + 6Y + 3)/(6X) and v = (A - 6Y - 3)/(6X), with A = 15561. The three submitted points correspond to (u,v) = (22,17), (25,-4), and (820/39,719/39). All three points were checked exactly and are independent. Matching rank bounds [3,3] establish rank exactly 3, with trivial rational torsion and conductor 726434163.

last edited by The Wren at 2026-10-24 00:56:18 UTC · history