Uniform bounds on the image of the arboreal Galois representations attached to non-CM elliptic curves

Let $\mathrm{\ell }$ be a prime number and let $F$ be a number field and $E\mathrm{/}F$ a non-CM elliptic curve with a point $\alpha \in E(F)$ of infinite order. Attached to the pair $(E,\alpha )$ is the $\mathrm{\ell }$-adic arboreal Galois representation $\omega_{E,\alpha,\ell^{\infty}} : {\rm Gal}(\overline{F}/F) \to $\mathbb{Z}${\ell}^{2} \rtimes {\rm GL}{2}($\mathbb{Z}${\ell})$ describing the action of $\mathrm{G}\mathrm{a}\mathrm{l}(\bar{F}\mathrm{/}F)$ on points ${\beta }_n$ so that $\ell^{n} \beta{n} = \alpha$\mathrm{.}Wegiveanexplicitboundontheindexoftheimageof$\omega_{E,\alpha,\ell^{\infty}}$dependingonhow$\ell$-divisiblethepoint$\alpha$is,andtheimageoftheordinary$\ell$-adic Galois representation. The image of ${\omega }_{E,\alpha ,{\mathrm{\ell }}^{\mathrm{\infty }}}$ is connected with the density of primes $\mathfrak{p}$ for which $\alpha \in E({\mathbb{F}}_{\mathfrak{p}})$ has order coprime to $\mathrm{\ell }$.