If $f$ is an elliptic function (wrt a given lattice), and $R=p/q$ a rational function, then is there a formula allowing to compute the degree of the elliptic function $R(f)$ in Terms of the degree of $f$ and the polynomial degrees of $p$ and $q$?
I would like to use this to prove that if $f$ is an elliptic function such that $\wp$ and $\wp'$ can be rationally expressed through $f$, then $f$ must have degree $1$ (i.e. cannot exist).