curve #4746

y² = x³ − 822361130107110767704859061075

a-invariants: [0, 0, 0, 0, -822361130107110767704859061075]
c-invariants: [0, 710520016412543703296998228768800]
rank (lower bound): ≥ 9
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 1096481506809481023606478748100 ☆ record for cube-sum curves, rank ≥ 9
discriminant (Δ): -292152021830371165222660402571504965191696593532757210627230000 ☆ record for cube-sum curves, rank ≥ 9
Faltings height: 10.6649 ☆ record for cube-sum curves, rank ≥ 9
naive height: 151.2871 ☆ record for cube-sum curves, rank ≥ 9
primes of bad reduction: 2, 3, 5, 37, 41, 53, 73, 1231, 4831
regulator: 2674487998201.6181363932607282119952336824943925291401021
submitted by: David Renshaw
submitted at: 2026-10-20 21:27:57 UTC
last updated: 2026-10-20 21:27:57 UTC

Witness: 9 independent points log in to add more points →

(11019158827671/49, 36576905616998197506/343)
(47820873392121/4900, -112287697349482469331/343000)
(37257970820479/2601, 193001002946826505858/132651)
(350304584915631/1369, -6556288770459356143446/50653)
(420506348004489/1444, 8622865328068735172037/54872)
(4637037899528481/183184, 307654190121945532796721/78402752)
(98802203484786969/620944, 31053146583013744704153597/489303872)
(22377124010073241/1299600, -3065936107754222340057539/1481544000)
(193664403326466091/16641, 85226509708085730338154136/2146689)

Commentary

Cube-sum curve x^3 + y^3 = 349043376293530 (Weierstrass model y^2 = x^3 - 432*349043376293530^2). 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 21:27:57 UTC · history