The generators of $5$-class group of some fields of degree 20 over $\mathbb{Q}$

Let $\mathrm{\Gamma }=\mathbb{Q}(\sqrt[5]{n})$ be a pure quintic field, where $n$ is a positive integer, $5^{th}$ power-free. Let $k_0=\mathbb{Q}({\zeta }_5)$ be the cyclotomic field containing a primitive $5^{th}$ root of unity ${\zeta }_5$, and $k=\mathrm{\Gamma }({\zeta }_5)$ be the normal closure of $\mathrm{\Gamma }$. Let $C_{k,5}$ be the $5$-component of the class group of k. The purpose of this paper is to determine generators of $C_{k,5}$, whenever it is of type $(5,5)$ and the rank of the group of ambiguous classes under the action of $Gal(k\mathrm{/}{k}_0)=⟨\sigma ⟩$ is $1$.