$(1^3+1)(2^3+1)\cdots (n^3+1)$ is not a cube

For a positive integer $n,$ define $$C_n=\prod _{k=1}^n(k^3+1)\mathrm{.}$$ In this paper we prove that there are no cubes in the integer sequence $C_n,n=1,2,\cdots \mathrm{.}$