$j$-invariant map (awaiting review)

The $\boldsymbol{j}$-invariant map map or $\boldsymbol{j}$-map of a modular curve $X_H$ is the morphism $X_H\to X(1)$ induced by the inclusion $H\le \GL_2(\widehat\Z)$.

When $X_H$ is isomorphic to $\mathbb P^1$ this map is specified as an element of $\mathbb \Q(t)$, and when $X_H$ is specified by a plane model $f(x,y,z)=0$, it is an element of $\mathbb \Q(x,z)$.