Structure of relative genus fields of cubic Kummer extensions

Let $N=\mathbb{Q}(\sqrt[3]{D},\zeta_3)$, where $D>1$ is a cube free positive integer, $K=\mathbb{Q}(\zeta_3)$ be the cyclotomic field containing a primitive cube root of unity $\zeta_3$, $f$ the conductor of the abelian extension $N/K$, and $N^{*}$ be the relative genus field of the Kummer extension $N/K$ with Galois group $G=\operatorname{Gal}(N/K)$. The aim of the present work is to find out all positive integers $D$ and conductors $f$ such that $\operatorname{Gal}\left(N^{*}/N\right)\cong\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z}$. This allows us to give rise, in our next paper [2], to new phenomena concerning the chain $\Theta=(\theta_i)_{i\in\mathbb{Z}}$ of \textit{lattice minima} in the underlying pure cubic subfield $L=\mathbb{Q}(\sqrt[3]{D})$ of $N$.Comment: 17 pages, 3 figures, 3 table