curve #5313

y² = x³ − 115506075

a-invariants: [0, 0, 0, 0, -115506075]
c-invariants: [0, 99797248800]
rank (lower bound): ≥ 4
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 37423968300
discriminant (Δ): -5763594252343230000 ☆ record for cube-sum curves, rank ≥ 4
Faltings height: 2.2787 ☆ record for cube-sum curves, rank ≥ 4
naive height: 50.6528 ☆ record for cube-sum curves, rank ≥ 4
primes of bad reduction: 2, 3, 5, 17, 73
regulator: 1055.594844700055855711794038938193356584524050
submitted by: David Renshaw
submitted at: 2026-10-21 13:03:32 UTC
last updated: 2026-10-21 13:03:32 UTC

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

(511, 4234)
(20659, 2969348)
(7081/4, 589621/8)
(178339/289, 53702888/4913)

Commentary

Cube-sum curve x^3 + y^3 = 111690 (k = 2*3^2*5*17*73; Weierstrass model y^2 = x^3 - 432k^2, minimal model shown). Found by an exhaustive scan of cube-free k <= 7*10^5: minimal-model naive height computed for every k, a Mestre-Nagao a_p sieve over primes p = 1 mod 3 to rank candidates, then PARI ellrank 2-descent. k divisible by 9 gains a factor 3^6 in the minimal model, so k = 111690 has smaller naive height than the classical smallest rank-4 k = 21691; it is the minimum naive height over all rank-4 k in the scanned range (any larger k has naive height > 57).

last edited by David Renshaw at 2026-10-21 13:03:32 UTC · history