curve #5318

y² + y = x³ − 123745519

a-invariants: [0, 0, 1, 0, -123745519]
c-invariants: [0, 106916128200]
rank (lower bound): ≥ 4
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 40093548075
discriminant (Δ): -6615195873425251875
Faltings height: 2.2902
naive height: 50.7906
primes of bad reduction: 3, 5, 7, 367
regulator: 375.2262883638559062809132336544274795569790777
submitted by: David Renshaw
submitted at: 2026-10-21 13:08:48 UTC
last updated: 2026-10-21 13:08:48 UTC

Witness: 4 independent points log in to add more points →

(2569/4, 95049/8)
(2569, 129734)
(2821, 149418)
(2391739/25, 3698883388/125)

Commentary

Cube-sum curve x^3 + y^3 = 115605 (Weierstrass model y^2 = x^3 - 432k^2, minimal model shown). Found in an exhaustive scan of cube-free k <= 7*10^5 by minimal-model naive height, with a Mestre-Nagao a_p sieve (p = 1 mod 3) and PARI ellrank 2-descent; companion to #5313 (k = 111690).

last edited by David Renshaw at 2026-10-21 13:08:48 UTC · history