Explicit description of maps $\phi_\alpha\in End(E)$ for a CM elliptic curve Let $E$ be an elliptic curve with complex multiplication. When $n\in \mathbb{Z}\hookrightarrow End(E)$, we can define its associated map as
$\phi_n: P\mapsto P+..._{n \text{ times}}+P$.
However when $\alpha\not\in \mathbb{Z}$ (but is an element of an imaginary quadratic field and of $End(E)$), I am unsure what the explicit description of $\phi_\alpha$, the associated  endomorphism of $E$, is.
Question: What is the explicit description of the map $\phi_\alpha$?

A possible construction I have in mind is utilizing the isomorphism $\phi: \mathbb{C}/L_E\simeq E$ to write
$\phi_\alpha(P)=(\phi\circ \alpha\circ \phi^{-1})(P)$
where $\alpha$ denotes multiplication by $\alpha$. I'm unsure if this is explicit, though, and would not know how to compute this map for specific examples.
 A: Let $$E:y^2=4x^3-g_2(L)x-g_3(L)$$ be a complex elliptic curve with CM by $\Bbb{Z}[\alpha]$ and isomorphic to the complex torus $\Bbb{C}/L$ (ie. $\alpha$ is an imaginary quadratic integer such that $\alpha L\subset L$), the isomorphism being $z+L\mapsto (\wp_L(z),\wp_L'(z))$.
On the complex torus side the CM endomorphism $z+L\mapsto \alpha z+L$ gives the pair of functions $$(\wp_L(\alpha z),\wp_L'(\alpha z))$$

*

*$\wp_L'(\alpha z)$ is the unique odd $\alpha^{-1} L$ periodic function with triple poles at $\alpha^{-1}L$ and $\wp_L'(\alpha z)\sim -2\alpha^{-3} z^{-3}$ as $z\to 0$.


*$\wp_L(\alpha z)$ is the unique $\alpha^{-1} L$ periodic function with double poles at $\alpha^{-1}L$ and $\wp_L'(\alpha z)= \alpha^{-2} z^{-2}+o(1)$ as $z\to 0$.
This implies that $$\wp_L'(\alpha z) = \alpha^{-3} \sum_{P\in \alpha^{-1}L/L} \wp_L'(z+P), \qquad\wp_L(\alpha z) =C+ \alpha^{-2} \sum_{P\in \alpha^{-1}L/L} \wp_L(z+P)$$
where $C=-\alpha^{-2}\sum_{P\in \alpha^{-1}L/L,P\ne L} \wp_L(P)$.
That's it. On the elliptic curve side $\alpha^{-1}L/L$ becomes an order $m=|\alpha|^2$ subgroup $H\subset E[m]$, and $(\wp_L(\alpha z),\wp_L'(\alpha z))$ becomes $$(x(\phi_\alpha),y(\phi_{\alpha}))= (C+ \alpha^{-2} \sum_{h\in H} x(.+h), \alpha^{-3} \sum_{h\in H} y(.+h))\tag{1}$$
where $C= -\alpha^{-2}\sum_{h\in H-O} x(h)$.
Note that the RHS are rational functions in $x,y$, $g_2(L),g_3(L)$ and the coordinates of elements of $H$.
At first we don't know $H$, so try each order $m$ subgroup of $E[m]$ and find if $(1)$ is an endomorphism, ie. if $y(\phi_\alpha)^2=4x(\phi_\alpha)^3-g_2(L)x(\phi_\alpha)-g_3(L)$.
