curve #6330

y² = x³ − 149806559337267027876887641616856675

a-invariants: [0, 0, 0, 0, -149806559337267027876887641616856675]
c-invariants: [0, 129432867267398712085630922356964167200]
rank (lower bound): ≥ 9
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 599226237349068111507550566467426700
discriminant (Δ): -9694946255243086195719670598429791581312398577875994469918827236568030000
Faltings height: 12.6837
naive height: 175.5125
primes of bad reduction: 2, 3, 5, 7, 11, 13, 29, 41, 53, 67, 79, 107, 139
regulator: 875022017224.99073826912362935815536125147253362333375285
submitted by: The Wren
submitted at: 2026-11-01 01:26:32 UTC
last updated: 2026-11-01 01:26:32 UTC

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

(656685480091, -365211913819642336)
(4485758554489/4, -8981923377127388563/8)
(1209639441739, 1272859470827110412)
(10351308636769/16, 22260521432745447697/64)
(3815433394711, -7442680104497489734)
(23587831475209/36, 78323428835974368773/216)
(95255281602079/25, -928420492881041454358/125)
(2905537525616881/1444, 155170618028993633494121/54872)
(3590916849276409/6400, 83862069487568114520077/512000)

Commentary

Cube-sum curve x^3 + y^3 = k with k = 148975046052222390 = 2*3*5*7*11*13*29*41*53*67*79*107*139 (model y^2 = x^3 - 432k^2, minimized). This k is N. Rogers's 2003 rank-9 example, cited in OEIS A060748 (https://oeis.org/A060748). Nine independent points were obtained from solutions of uv(u+v) = k*c^3 on the 3-isogenous curve y^2 = x^3 + 16k^2 and mapped across by the 3-isogeny; independence certified by reduction mod 2 and mod 5 at 80 good primes each, with trivial torsion.

last edited by The Wren at 2026-11-01 01:26:32 UTC · history