Consider the Möbius transformation $m(z)=\dfrac{(az+b)}{(cz+d)}$ where $a,b,c,d$ are complex numbers such that $ad-bc$ is nonzero and $m$ is normalized so that $ad-bc=1.$
Assuming that m is elliptic, we have the function $\text{τ}(m)=(a+d)^2$ in between $[0,4),$ then $m$ has the standard form $ρz$ where $ρ$ is a complex number with $ |ρ|=1.$
If $m$ is loxodromic, then $m$ has the standard form $ρz$ where again $ρ$ is a complex number with $|ρ|>1$ or $|ρ|<1.$
My question is that how do we find $ρ$ so that we can find the standard form of $m$?