Let $\Gamma\subset {SL}_2(\mathbb Z)$ be a congruence subgroup, and let $X(\Gamma)$ be the associated compact modular curve. The inclusion $\Gamma \subset SL_2(\mathbb Z)$ induces a canonical projection $p:X(\Gamma)\to X(1)$ which is a non-constant holomorphic map of compact Riemann surfaces, where $X(1)=X(SL_2(\mathbb Z))$. The map $p$ ramifies at most over the elliptic points $i=\sqrt{-1}$ and $\mu_3=e^{2\pi i/3}$ of $X(1)$ and over the cusp $\infty$ of $X(1)$. Let $d$ be the degree of $p$ and let $\overline{\mathbb Q}$ be an algebraic closure of $\mathbb Q$.

Assumption $(*)$: There exists a smooth, projective and connected curve $X(\Gamma)_{\overline{\mathbb Q}}$ over $\overline{\mathbb Q}$ such that its base change to $\mathbb C$ corresponds to $X(\Gamma)$.

If assumption $(*)$ is satisfied, does there exists a non-constant morphism $X(\Gamma)_{\overline{\mathbb Q}}\to \mathbb P^1$ of algebraic curves over $\bar{\mathbb Q}$ of degree $d$, which ramifies at most over $i$, $\mu_3$ and $\infty$ (viewed as points of $\mathbb P^1=X(1)_{\overline{\mathbb Q}}$)?

Is assumption $(*)$ always satisfied?