curve #5321

y² = x³ − 172596675

a-invariants: [0, 0, 0, 0, -172596675]
c-invariants: [0, 149123527200]
rank (lower bound): ≥ 4
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 55921322700
discriminant (Δ): -12869112479496030000
Faltings height: 2.3457
naive height: 51.4561
primes of bad reduction: 2, 3, 5, 37, 41
regulator: 1244.172705185208202699298063703193165236293190
submitted by: David Renshaw
submitted at: 2026-10-21 13:09:03 UTC
last updated: 2026-10-21 13:09:03 UTC

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

(1591, 62086)
(33559, 6147698)
(4921/4, 328819/8)
(13801/4, 1617899/8)

Commentary

Cube-sum curve x^3 + y^3 = 136530 (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 13:09:03 UTC · history