Let $E$ be an elliptic curve with complex multiplication and let $F=\mathrm{End}(E)\otimes \mathbb{Q}$. Suppose $End(E) \cong \mathcal{O}$ where $\mathcal{O}$ is the order of conductor $f$ in $F$.
Is there any relationship between the minimal discriminant of $E$ and the field discriminant of $F/\mathbb{Q}$?