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Let $E/\mathbb{Q}$ be an elliptic curve and $p \in \{5,7,11 \}$ be a prime.We determine the possibilities for $E(\mathbb{Q}(\zeta_{p}))_{tors}$.Additionally, we determine all the possibilities for$E(\mathbb{Q}(\zeta_{16}))_{tors}$ and $E(\mathbb{Q}(\zeta_{27}))_{tors}$.Using these results we are able to determine the possibilities for$E(\mathbb{Q}(\mu_{p^{\infty}}))_{tors}$.

Torsion groups of elliptic curves over $\mathbb{Q}(\mu_{p^{\infty}})$