  The moduli space of degree $d$ morphisms on $\mathbb{P}^1$ has received much
study. McMullen showed that, except for certain families of Latt\`es maps,
there is a finite-to-one correspondence (over $\mathbb{C}$) between classes of
morphisms in the moduli space and the multipliers of the periodic points. For
degree 2 morphisms Milnor (over $\mathbb{C}$) and Silverman (over $\mathbb{Z}$)
showed that the correspondence is an isomorphism. In this article we address
two cases: polynomial maps of any degree and rational maps of degree 3.
