On possibilities of 3-adic Galois images associated to isogeny-torsion graphs

Let $E$ be a non CM elliptic curve defined over $\text{Q}$. There is an isogeny-torsion graph associated to $E$ and there is also a Galois representation ${\rho }_{E,l^{\mathrm{\infty }}}\text{\hspace{0.17em}⁣}:\text{Gal}(\text{Qbar}\mathrm{/}\text{Q})\to {\text{GL}}_2({\text{ZZ}}_{\mathrm{\ell }})$ associated to $E$ for every prime $\mathrm{\ell }\mathrm{.}$ In this article, we explore the relation between these two objects when $\mathrm{\ell }=3$. More precisely, we give a classification of 3-adic Galois images associated to vertices of the isogeny-torsion graph of $E\mathrm{.}$