  In this paper, we introduce the study of minimal torsion curves within a
fixed geometric isogeny class. For a $\overline{\mathbb{Q}}$-isogeny class
$\mathcal{E}$ of elliptic curves and $N \in \mathbb{Z}^+$, we wish to determine
the least degree of a point on the modular curve $X_1(N)$ associated to any $E
\in \mathcal{E}$. In the present work, we consider the cases where
$\mathcal{E}$ is rational, i.e., contains an elliptic curve with rational
$j$-invariant, or where $\mathcal{E}$ consists of elliptic curves with complex
multiplication (CM). If $N=\ell^k$ is a power of a single prime, we give a
complete characterization upon restricting to points of odd degree, and also in
the case where $\mathcal{E}$ is CM. We include various partial results in the
more general setting.
