Imaginary quadratic Minkowski disk bound

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