Minimal torsion curves in geometric isogeny classes

In this paper, we introduce the study of minimal torsion curves within a fixed geometric isogeny class. For a Q̄-isogeny class E of elliptic curves and N ∈ Z⁺, we wish to determine the least degree of a point on the modular curve X₁(N) associated to any E ∈ E. In the present work, we consider the cases where E is rational, i.e., contains an elliptic curve with rational j-invariant, or where E consists of elliptic curves with complex multiplication (CM). If N=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 E is CM. We include various partial results in the more general setting.