Let $K$ be a quadratic imaginary field. To simplify my life, let us assume that $K$ has class number one.

Consider the following infinite set:

$S_1:=$ $\{$ $E\subseteq\mathbf{P}^2(\mathbf{C})$ is an embedded elliptic curve defined over $\mathcal{O}_K$ with global minimal Weierstrass equation and with CM by $K$ }

Let $S_2=\{N_{K/\mathbf{Q}}(f_E):E\in S_1 \mbox{where $f_E\subseteq\mathcal{O}_K$ is the conductor of $E/\mathcal{O}_K$ }\}$.

Q1: Is it possible to find an explicit expression $C_K$ (where we think of $K$ as a parameter; forget for the moment that there are only 9 such fields), depending only on $K$ (involving for example the discriminant of $K$), such that there exists always an elliptic curve $E\in S_1$ such that $N_{K/\mathbf{Q}}(f_E)\leq C_K$?

Q2: What do we know about the minimal values in the set $S_2$ ?