curve #5319

y² = x³ − 135273675

a-invariants: [0, 0, 0, 0, -135273675]
c-invariants: [0, 116876455200]
rank (lower bound): ≥ 4
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 43828670700
discriminant (Δ): -7905153807938430000
Faltings height: 2.3051
naive height: 50.9688
primes of bad reduction: 2, 3, 5, 17, 79
regulator: 1213.169027749376293729471024560155780107618126
submitted by: David Renshaw
submitted at: 2026-10-21 13:08:53 UTC
last updated: 2026-10-21 13:08:53 UTC

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

(4819, 334328)
(2449/4, 77657/8)
(14299/25, 899968/125)
(413881/400, 249477971/8000)

Commentary

Cube-sum curve x^3 + y^3 = 120870 (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 13:08:53 UTC · history