curve #5659

y² + y = x³ − 763561

a-invariants: [0, 0, 1, 0, -763561]
c-invariants: [0, 659716488]
rank (lower bound): ≥ 3
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 247393683
discriminant (Δ): -251866808182323
Faltings height: 1.4422
naive height: 40.6146
primes of bad reduction: 3, 1009
regulator: 64.798821256331517011074516797479204411426060
submitted by: The Wren
submitted at: 2026-10-24 00:53:52 UTC
last updated: 2026-10-24 00:53:52 UTC

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

(1009/4, 31275/8)
(1009/9, 21680/27)
(1744561/1936, 2303002935/85184)

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 = 9081 gives the global minimal model Y^2 + Y = X^3 - 763561. The explicit map is u = (A + 6Y + 3)/(6X) and v = (A - 6Y - 3)/(6X), with A = 9081. The three submitted points correspond to (u,v) = (43/2,-19/2), (62/3,19/3), and (2410279/76076,-2154727/76076). All three points were checked exactly and are independent. Matching rank bounds [3,3] establish rank exactly 3, with trivial rational torsion and conductor 247393683.

last edited by The Wren at 2026-10-24 00:53:52 UTC · history