The moduli space of $G$-algebras

Let $L$ be a Galois algebra with Galois group $G$ and let $x$ be a normal element of $L$. The moduli space $\mathcal{X}$ of pairs $(L,x)$ is isomorphic to an open subset of the quotient variety $\mathbb{P}\mathrm{/}G$, where $\mathbb{P}$ is the projective space of the regular representation of $G$. We provide a formula for the height of any pair $(L,x)\in \mathcal{X}(\mathbb{Q})$ in terms of algebraic invariants of $L$ and $x$ with respect to a natural adelic metric on the anticanonical divisor of $\mathbb{P}\mathrm{/}G$.