On Possible Genus 0 adelic Galois Images of non CM Elliptic Curves over $\mathbb{Q}$

Let $E$ be an elliptic curve defined over $\mathbb{Q}$/ Associated to $E$, there is an adelic Galois representation ${\rho }_E\text{\hspace{0.17em}⁣}:\mathrm{G}\mathrm{a}\mathrm{l}(\bar{\mathbb{Q}}\mathrm{/}\mathbb{Q})\to {\mathrm{G}\mathrm{L}}_2(\widehat{\mathbb{Z}})$. In this article, we give possibilities for groups generated by ${\rho }_E(\mathrm{G}\mathrm{a}\mathrm{l}(\bar{\mathbb{Q}}\mathrm{/}\mathbb{Q}))$ and $-I$ as $E$ varies over non CM elliptic curves defined over $\mathbb{Q}$ such that the group $\pm {\rho }_E(\mathrm{G}\mathrm{a}\mathrm{l}(\bar{\mathbb{Q}}\mathrm{/}\mathbb{Q}))$ has genus 0. However, our list is not minimal.