Let $E : y^2 = x^3 + ax + b$ be an elliptic curve over the field $K$, char $K \ne 0$.
We know that the differential $\omega = \frac{dx}{y}$ is holomorphic in infinity because we can write it as $\frac{2 dy}{3x^2 + a}$.
I can't understand, though, how $x \omega$ has a double pole at infinity. I can't make sense of it.