Shapes of tame abelian number fields

The shape of a number field $K$ of degree $n\ge 2$ is the similarity class of the lattice obtained by projecting ${\mathcal{O}}_K$ orthogonally away from the rational line in Minkowski space. It defines a point in the space of shapes of rank-$(n-1)$ lattices, ${\mathcal{S}}_{n-1}:={\mathrm{G}\mathrm{L}}_{n-1}(\mathbb{Z})\mathrm{\backslash }{\mathrm{G}\mathrm{L}}_{n-1}(\mathbb{R})\mathrm{/}{\mathrm{G}\mathrm{O}}_{n-1}(\mathbb{R})$, where ${\mathrm{G}\mathrm{O}}_{n-1}(\mathbb{R})$ is the group of orthogonal similitudes. Let $G$ be a fixed nontrivial finite abelian group, and let $K\mathrm{/}\mathbb{Q}$ vary over tamely ramified Galois extensions with Galois group isomorphic to $G$. We study the variation of the shapes of such number fields in ${\mathcal{S}}_{n-1}$, where $n=\mathrm{\mid }G\mathrm{\mid }$.