Let $E/\mathbb{Q}$ be an elliptic curve with CM, with the endomorphism ring $R=\mathrm{End}_{\overline{\mathbb{Q}}}(E)$. Then for any integer $m$, we have the mod-$m$ Galois representation $\rho_m:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \hookrightarrow (R/mR)^{*}$. I would like to understand the image. Page 12 in a master thesis says the image has a bounded index (in $(R/mR)^{*}$) when $m$ is large enough, but I couldn't find any other reference for this. If it is true, can anyone provide a suitable reference?
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