Consider the isomorphism $\Phi:\mathbb{C}/L\rightarrow E/\mathbb{C}$ given by $z\mapsto (\wp(z),\wp'(z))$ (when $z\not\in L$, and $z\mapsto 0$ if $z\in L$). In particular,
$$\wp(z)=\sum_{l\in L^\times} \frac{1}{(z-l)^2}-\frac{1}{l^2}$$
Are there any known results on (algorithms for?) finding which specific complex points $z$ satisfy $\Phi(z)\in E(\mathbb{Q})$? That is, can we classify explicitly which (/examples of) $z\in \mathbb{C}/L$ map to rational points of the elliptic curve?
Is this straightforward and if not is there literature on it? One can substitute $\mathbb{Q}$ for some extension $\mathbb{Q}[\sqrt{-d}]$ and similarly ask when $\Phi(z)\in E(\mathbb{Q}[\sqrt{-d}])$.