curve #4710

y² + y = x³ − 8840484855150716167

a-invariants: [0, 0, 1, 0, -8840484855150716167]
c-invariants: [0, 7638178914850218768072]
rank (lower bound): ≥ 7
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 35361939420602864667 ☆ record for cube-sum curves, rank ≥ 7
discriminant (Δ): -33762602508832445329787086586341359564003 ☆ record for cube-sum curves, rank ≥ 7
Faltings height: 6.4556 ☆ record for cube-sum curves, rank ≥ 7
naive height: 100.7749 ☆ record for cube-sum curves, rank ≥ 7
primes of bad reduction: 3, 13, 19, 41, 139, 271
regulator: 6092151.334835195171137323313228863378826233920433510
submitted by: David Renshaw
submitted at: 2026-10-20 21:10:55 UTC
last updated: 2026-10-20 21:10:55 UTC

Witness: 7 independent points log in to add more points →

(2075047, 307073487)
(8496097, 24585339702)
(53376973/9, 381617199359/27)
(207062323/81, 2044397401091/729)
(2276676691/1089, 19580521361716/35937)
(25210091257/6724, 3651667342265937/551368)
(25409887633/11664, 1541904778397159/1259712)

Commentary

Cube-sum curve x^3 + y^3 = 1144421889 (Weierstrass model y^2 = x^3 - 432*1144421889^2). This A is the rank-7 example of Elkies-Rogers (2004), Elliptic Curves x^3+y^3=k of High Rank, https://arxiv.org/abs/math/0403116 ; its 3-isogenous Mordell model is curve#2923. Rediscovered here as the smallest rank-7 A in a Mestre-Nagao sieve over all A <= 10^12.

last edited by David Renshaw at 2026-10-20 21:10:55 UTC · history