Let $K$ be a quadratic imaginary field. Let $L$ be a number field which contains $K$ and let $E/L$ be an elliptic curve defined over $L$ with complex multiplication by $K$, i.e. such that $End\_{\overline{L}}(E)\otimes\_{\mathbf{Z}}\mathbf{Q}\simeq K$. One may associate to $E$ a Groessencharacter of $L$
taking values in $K$
$$
\psi:\mathbb{A}\_L^{\times}/L^{\times}\rightarrow K^{\times}.
$$, 


Q1: Is there an obvious upper bound for the conductor of $\psi$ depending only on $L$ (which seems, a priori, to be almost equivalent to ask for an upper bound for the conductor of $E/L$)?


Q2: If $K$ has class number one, then we may choose $L=K$. Can we give an explicit formula for the conductor of $\psi$ in this very special situation? 



On the conductor of the Groessencharacter of a CM elliptic curve
The explicit formula should be: For each prime $p$ of $L$, take the local ring $\mathcal O\_p$ of $p$ and look at the induced map $\mathcal O\_p^{\times} \to \mathbb A\_L^\times \to K^\times$. Let $\pi$ be a uniformizer and take the smallest $n$ such that the image of $1+ \pi^n \mathcal O\_p$ along this map is trivial, then take $\# \mathcal O\_p /\pi^n$. 


This is the local factor, then you take the product over all $p$.


This follows from class field theory.


