Let $F$ be a number field and $\mathcal{O}$ an order in an imaginary quadratic field $K$. Assume $K\subseteq F$. In Lang's *Elliptic Functions*, it is shown that over that there is a bijection between, the isomorphism classes of elliptic curves over $\bar{F}$ and the class group of $\mathcal{O}$. This is done by defining an action the class group on lattices of $\mathbb{C}$. Moreover, any two elliptic curves $E,E'$ over $\bar{F}$ that have CM by $\mathcal{O}$ are isogenous over $\bar{F}$

However, I'm told this can be done F-rationally and in fact this is done by Deuring. But I don't read German. This is stated in Products of CM Elliptic Curves (equation 55, pg.20), but the proof is in service of much more complicated results. $$\{E'/F\; |\; E' \mbox{ is $F$-isogenous to $E$ and } \mathrm{End}(E)\cong \mathrm{End}(E') \} \cong \mathrm{Cl}(\mathcal{O})$$ I'm wondering if there is a simpler proof in the vein of defining an action of the class group on lattices of $F$ and if anyone can give me a reference.

More precisely, for an elliptic curve $E/F$ with CM by $\mathcal{O}$, let $E=E_1,\dots, E_h$ denote representatives for the $\bar{F}$ isomorphism classes of elliptic curves with CM by $\mathcal{O}$. For $j=2\dots h$, I want to find an elliptic curve $E'/F$ such that $E'$ is in the $\bar{F}$-isomorphism class of $E_j$ and $E'$ is isogenous to E over $F$.

isomorphic(not just isogenous). Tracking $F$-rationality through elliptic functions is a bit involved. $\endgroup$ – nfdc23 Aug 10 '17 at 2:33