curve #4742

y² + y = x³ − 5587779175777

a-invariants: [0, 0, 1, 0, -5587779175777]
c-invariants: [0, 4827841207871112]
rank (lower bound): ≥ 6
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 1810440452951667 ☆ record for cube-sum curves, rank ≥ 6
discriminant (Δ): -13488455282649535687953243123 ☆ record for cube-sum curves, rank ≥ 6
Faltings height: 4.0765 ☆ record for cube-sum curves, rank ≥ 6
naive height: 72.2264 ☆ record for cube-sum curves, rank ≥ 6
primes of bad reduction: 3, 17, 307, 523
regulator: 30232939.358748973823233310122636666489107144304934
submitted by: David Renshaw
submitted at: 2026-10-20 21:27:41 UTC
last updated: 2026-10-20 21:27:41 UTC

Witness: 6 independent points log in to add more points →

(24827977/576, 119318180923/13824)
(53228974753/2862864, -4438835273216879/4843965888)
(76086207361/3348900, 15185638639662859/6128487000)
(7043625331417/114190596, 18469781212844538377/1220240708856)
(6672158799337/175986756, 16327054300881132287/2334640305096)
(47901530520433/1742895504, 283423710305398716649/72762401500992)

Commentary

Cube-sum curve x^3 + y^3 = 24565833 (Weierstrass model y^2 = x^3 - 432*24565833^2). Source: Elkies & Rogers, 'Elliptic Curves x^3+y^3=k of High Rank', ANTS-VI (LNCS 3076, 2004), arXiv:math/0403116, https://arxiv.org/abs/math/0403116 Points mapped from the 3-isogenous curve uv(u+v)=A given there via the dual isogeny.

last edited by David Renshaw at 2026-10-20 21:27:41 UTC · history