Consider an elliptic curve $E/\Bbb Q$ and let $\Phi:\Gamma_0(N)\backslash\overline{\mathfrak{H}}\rightarrow E(\Bbb C)$ be the analytical description of its modular parametrization. We know that this is a map defined over $\Bbb Q$ between algebraic curves over $\Bbb Q$, but in general for $\tau\in{\mathfrak{H}}$ the field of definition of the corresponding point in $X_0(\Bbb C)$ is not easy to identify.

Is it possible to identify points in $\mathfrak H$ that correspond to real points of $X_0(N)$ or points that map to real points of $E$?

Can anything be said about points mapping to real points under the Weierstrass parametrization $(\wp,\wp'):\Bbb C/\Lambda\rightarrow E(\Bbb C)$?

Thanks