Imaginary quadratic Minkowski disk bound
= Imaginary quadratic Minkowski disk bound
Let $K=\mathbb Q(\sqrt{-d})$ for square-free $d>0$ with $d\not\equiv3\pmod4$, 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
$$
0<|N_{K/\mathbb Q}(\alpha)|
\le\frac{4\sqrt d}{\pi}N(I).
$$