%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{The groups that occur}\label{sec:occur} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Let $\mathcal G_{10}$ be the set of the $69$ groups listed in \Cref{thm:main}. In this section we show that $65$ of them are torsion groups of elliptic curves over number fields of degree $10$: $38$ of them already occur in degree $1$, $2$ or $5$ (\Cref{subsec:base}), $15$ occur because the corresponding modular curve has a function of the right degree (\Cref{subsec:families}), and for the remaining $12$ we give explicit elliptic curves (\Cref{subsec:explicit}). The remaining $4$ groups $\tg{2}{26}$, $\tg{3}{15}$, $\tg{3}{18}$ and $\tg{6}{12}$ are realized in \Cref{sec:sporadic}, where we also determine all the elliptic curves with these torsion groups. \subsection{Groups from smaller degrees}\label{subsec:base} The following lemma is well known. We give the proof, which uses the same trick as \cite[Lemma 3.1]{Sd}. \begin{lemma}\label{lem:base} Let $F$ be a number field of degree $e$, let $E/F$ be an elliptic curve and let $d$ be a multiple of $e$. Then there is a number field $K\supseteq F$ of degree $d$ such that $E(K)_\tors=E(F)_\tors$. In particular, $\Phi(e)\subseteq\Phi(d)$. \end{lemma} \begin{proof} By Merel's theorem \cite{merel} there are only finitely many points $P\in E(\overline F)_\tors$ with $[F(P):F]\leq d/e$. Let $M$ be the Galois closure over $F$ of the compositum of the fields $F(P)$ for these points $P$, let $\ell$ be a rational prime which is unramified in $M$, and let $\alpha$ be a root of $x^{d/e}-\ell$. This polynomial is Eisenstein at every prime of $F$ above $\ell$, so $K=F(\alpha)$ has degree $d/e$ over $F$, and every prime of $F$ above $\ell$ is totally ramified in $K$. If $P\in E(K)_\tors$, then $F(P)\subseteq K\cap M$, and $K\cap M$ is an extension of $F$ which is both unramified and totally ramified at the primes above $\ell$. Hence $K\cap M=F$ and $P\in E(F)$. \end{proof} By Mazur's theorem \cite{mazur}, the theorem of Kamienny \cite{kamienny} and Kenku and Momose \cite{KM}, and \cite[Theorem 1.1]{N5}, the sets $\Phi(1)$, $\Phi(2)$ and $\Phi(5)$ consist of the groups \begin{alignat*}{2} \Phi(1):\ &\torz{n},\ n=1,\dots,10,12, &&\qquad\tg{2}{2n},\ n=1,\dots,4,\\ \Phi(2):\ &\torz{n},\ n=1,\dots,16,18, &&\qquad\tg{2}{2n},\ n=1,\dots,6,\\ &\tg33,\ \tg36,\ \tg44, && \\ \Phi(5):\ &\torz{n},\ n=1,\dots,22,24,25,28,30, &&\qquad\tg{2}{2n},\ n=1,\dots,9. \end{alignat*} By \Cref{lem:base} all these groups lie in $\Phi(10)$. There are $38$ of them: the groups $\torz n$ for $1\leq n\leq 25$, $n\neq 23$, and $n\in\{28,30\}$, the groups $\tg{2}{2n}$ for $1\leq n\leq 9$, and $\tg33$, $\tg36$ and $\tg44$. \subsection{Groups from functions on modular curves}\label{subsec:families} \begin{lemma}\label{lem:hilbert} Let $m\in\{1,2,3,4,6\}$, let $m\mid n$ with $n\geq5$, put $F=\Q(\zeta_m)$, and suppose that $X_1(m,n)$ has a function of degree $d_m=10/\varphi(m)$ over $F$. Then there are infinitely many pairs $(K,E)$, consisting of a number field $K$ of degree $10$ and an elliptic curve $E/K$ with $E(K)_\tors\simeq\tg mn$, such that the $j$-invariants $j(E)$ are pairwise distinct. \end{lemma} \begin{proof} This is \cite[Lemma 5.1]{DS}. The idea is to apply Hilbert's irreducibility theorem to $f$ and to its compositions with the forgetful maps from the modular curves with larger level structures, which have degree at least $2$, so that for infinitely many fibres of $f$ the elliptic curve gains no further torsion over the residue field. \end{proof} We use the following model of $X_1(m,n)$, as in Derickx's package \texttt{mdsage} \cite{mdsage}. Let $H\subseteq(\torz{mn})^\times$ be the subgroup generated by $-1$ and by $1+kn$ for $1\leq k