It is known that, given a number field $K$ with degree $n$, one can embed the corresponding algebraic integers $O_K$ in $\mathbb{R}^n$ as a lattice $\Lambda_K$. From here on, one can construct the torus $\mathbf{T}_K := \mathbb{R}^n/\Lambda_K$.
My question: are there any results relating $K$ and $\mathbf{T}_K$? In particular, I'm looking for results that are not (obviously) deduced from considering $\Lambda_K$ only.