Minkowski convex body theorem

If $\Lambda \subset {\mathbb{R}}^n$ is a full lattice and a measurable convex centrally symmetric set $C$ has volume greater than $2^n\mathrm{covol}(\Lambda )$, then $C$ contains a nonzero lattice point.

For a real quadratic field $K$ with embeddings ${\sigma }_1,{\sigma }_2$, the imageis a lattice in ${\mathbb{R}}^2$ of covolume $\sqrt{|d_K|}$, and the product of the two coordinates is the field norm.

Let $K=\mathbb{Q}(\sqrt{-d})$ for square-free $d>0$ with $d\not\equiv 3(\mathrm{mod}4)$, so ${\mathcal{O}}_K=\mathbb{Z}[\sqrt{-d}]$. An integral ideal $I$ is a lattice in $\mathbb{C}\simeq {\mathbb{R}}^2$ of covolume $\sqrt{d}N(I)$. Applying the Minkowski convex body theorem to a disk gives a nonzero $\alpha \in I$ satisfying