Rational points on modular curves via maps to elliptic curves with rank zero

A fundamental problem in arithmetic geometry is to determine the image of the mod $N$ Galois representation for all elliptic curves over $\mathbb{Q}$ and integers $N\ge 1$. For a given subgroup $G\le {\mathrm{G}\mathrm{L}}_2(\mathbb{Z}\mathrm{/}N\mathbb{Z})$, there is a modular curve $X_G$ whose rational points parametrize elliptic curves for which the image of the mod $N$ Galois representation is contained in $G$. If $X_G$ admits a map to an elliptic curve $E\mathrm{/}\mathbb{Q}$ for which $E(\mathbb{Q})$ has rank $0$, then its rational points can be effectively determined, provided that a map $X_G\to E$ is known. In this article, we give a method for constructing such maps. Using this method, together with existing methods and results, we systematically determine the rational points of $X_G$ for more than $99\mathrm{\%}$ of modular curves of level at most $70$.