Let $E$ be an elliptic curve over $\mathbb C$ with CM by ring of integers $O_K$ of an imaginary quadratic number field $K$. Let $O$ be an order of $O_K$.

Is there a number field $L$ such that $E$ has a model $E_L$ over $L$ with $\mathrm{End}_L(E_L) = O$?

Of course, if $O = O_K$, the answer is positive. But what if $O$ has a non-trivial conductor?

Concretely: Let $E : y^2= x^3 +x$ over $\mathbb Q$, and let $f\geq 2$ be an integer. Is there a number field $L$ such that $\mathrm{End}_L(E_L) = \mathbb Z[fi]$?