The minimal polynomial of $\theta$ over $\mathbb{Z}$ is the depressed cubic

Let $D,E\in\mathbb{Z}$ with $x^3+Dx+E$ irreducible over $\mathbb{Q}$, and let $\theta\in\mathcal{O}_K$ be the root of that cubic in $K=\mathbb{Q}[x]/(x^3+Dx+E)$. Then the minimal polynomial of $\theta$ over $\mathbb{Z}$ is exactly $X^3+DX+E$ in $\mathbb{Z}[X]$.

This identifies the integral model, which is what lets Kummer-Dedekind's factorisation of $p$ be read off from the factorisation of $X^3+DX+E$ modulo $p$.