Counting metacyclic fields

By a metacyclic field $K$ we mean the Galois closure of a pure field $\mathbb{Q}(\sqrt[\mathrm{\ell }]{D})$, $D\in \mathbb{Z}$, of odd prime degree $\mathrm{\ell }$. Let ${\mathcal{M}}_{\mathrm{\ell }}$ denote the collection of isomorphism classes of metacyclic fields of degree $\mathrm{\ell }(\mathrm{\ell }-1)$. Write $N_{\mathrm{\ell }}(X)=\mathrm{\#}\{K\in {\mathcal{M}}_{\mathrm{\ell }}:\mathrm{\mid }{\mathrm{\Delta }}_K\mathrm{\mid }\le X\}$ for the associated counting function, where ${\mathrm{\Delta }}_K$ denotes the discriminant of $K$. We show $$N_{\mathrm{\ell }}(X)\sim A_{\mathrm{\ell }}{X}^{\frac{1}{(\mathrm{\ell }-1{)}^2}}(\log X{)}^{\mathrm{\ell }-2},$$ for an explicit constant $A_{\mathrm{\ell }}$. We express $A_{\mathrm{\ell }}$ as a rational number times a product over primes of a degree $\mathrm{\ell }$ polynomial in $1\mathrm{/}p$.