Let $K$ be an imaginary quadratic field (suppose even that it has class number $1$) with ring of integers $\mathcal{O}_K$ and consider an integral ideal $\mathfrak{f} \lhd \mathcal{O}_K$.
Is it true that there exists an elliptic curve $E$, an invariant differential $\omega$ and a $z_0 \in \mathbb{C}$ such that $$ E \cong \mathbb{C}/(z_0 \cdot \mathfrak{f}) $$
Where $z_0 \cdot \mathfrak{f}$ is the period lattice of $\omega$ ? If it is, can anyone explain how or point me out to some reference ?
Thank you.