So I've heard in passing that for any congruence modular curve $X$ (over $\mathbb{C}$), there is a $g\in\text{GL}_2(\mathbb{Q})^+$ such that $X$ is birational to a plane curve in $\mathbb{C}^2$ given by the subset $\{(j(\tau),j(g(\tau))) : \tau\in\mathbb{H}\}$

In other words, for every $g\in\text{GL}_2(\mathbb{Q})^+$, the two functions $j,j\circ g: \mathbb{H}\rightarrow\mathbb{C}$ satisfy some polynomial relation $\Phi_g(j,j\circ g)$ (with $\mathbb{C}$-coefficients) such that $\text{Spec }\mathbb{C}[x,y]/\Phi_g(x,y)$ is isomorphic to some affine congruence modular curve, and every congruence modular curve can be obtained this way via some $g\in\text{GL}_2(\mathbb{Q})^+$.

A calculation shows that if $g\in\text{GL}_2(\frac{1}{n}\mathbb{Z})^+$, then $g$ conjugates the principal congruence subgroup $\Gamma(n^2)$ into $\text{SL}_2(\mathbb{Z})$, so $j\circ g$ will be invariant under $\Gamma(n^2)$, so $j,j\circ g$ will generate the function field of some congruence modular curve of level dividing $n^2$. It seems that the corresponding congruence subgroup is then just the largest subgroup of $\text{SL}_2(\mathbb{Z})$ which $g$ conjugates into $\text{SL}_2(\mathbb{Z})$.

Is it true that every congruence modular curve can be obtained in this way? (Is this described anywhere?)

If $K$ is the number field obtained by adjoining the coefficients of $\Phi_g(x,y)$ to $\mathbb{Q}$, then is $\text{Spec }K[x,y]/\Phi_g(x,y)$ the the usual arithmetic model of the modular curve? (e.g., for some $g$ so that $\Phi_g$ gives a principal congruence modular curve $Y(n)$, is $K = \mathbb{Q}$ or $\mathbb{Q}(\zeta_n)$?)