A Classification of Genus 0 Modular Curves with Rational Points

Let $E$ be a non-CM elliptic curve defined over $\mathbb{Q}$. Fix an algebraic closure $\bar{\mathbb{Q}}$ of $\mathbb{Q}$. We get a Galois representation $${\rho }_E\text{\hspace{0.17em}⁣}:Gal(\bar{\mathbb{Q}}\mathrm{/}\mathbb{Q})\to G{L}_2(\widehat{\mathbb{Z}})$$ associated to $E$ by choosing a compatible bases for the $N$-torsion subgroups of $E(\bar{\mathbb{Q}})\mathrm{.}$ Associated to an open subgroup $G$ of $G{L}_2(\widehat{\mathbb{Z}})$ satisfying $-I\in G$ and $det(G)={\widehat{\mathbb{Z}}}^{\times }$, we have the modular curve $(X_G,{\pi }_G)$ over $\mathbb{Q}$ which loosely parametrises elliptic curves $E$ such that the image of ${\rho }_E$ is conjugate to a subgroup of $G^t\mathrm{.}$ In this article we give a complete classification of all such genus $0$ modular curves that have a rational point. This classification is given in finitely many families. Moreover, each such modular curve can be explicitly computed.