Let $E_1,E_2$ be two elliptic curves over a field $k$ of characteristic zero with complex multiplication. Let $R_1,R_2$ be the endomorphism rings of the elliptic curves and suppose that the fraction fields of $R_1,R_2$ are equal. Is it necessarily the case that $E_1,E_2$ are isogenous?
Let $E$ be an elliptic curve over a field $k$ of characteristic $0$ with complex multiplication by $R$. Let $K$ be the fraction field of $R$ and let $S$ be the ring of integers in $R$. Can we construct an elliptic curve $E'$ with complex multiplication by $S$ which is isogenic to $E$?
Can we do this over $k$?
This is all clear if $k = \mathbb C$ but what about the general case? I am most interested in $k = \mathbb Q$. Perhaps you can do a Lefschetz style proof but I would rather not.