As we know, there are three types of differential forms on an elliptic curve: those of first kind -- which have no poles or zeroes, those of the second kind -- which have poles with residue 0, and third kind -- which have simple poles with non-zero residue.
Now, if the elliptic curve is given in Weierstrass form I know that $\frac{dx}{y}$ is a differential of the first kind, and that $\frac{xdx}{y}$ is a differential of the second kind, and one could similarly write a differential of the third kind.
However, I was wondering if there is a way of determining the type of a meromorphic differential form, say $\frac{u(x_1,x_2)dx_1}{f(x_1,x_2)}$, where $f$ is a cubic polynomial in the variables $x_1,x_2$, and $u(x_1,x_2)$ is any polynomial, without changing variables to get $f$ into Weirestrass form?