The matter of explicitly finding the order of a rational function on an elliptic curve in the projective plane at infinity (i.e. at the point $(0, 1, 0)$) still seems unclear.
For example, Silverman (in The Arithmetic of Elliptic Curves) states that the order of the rational function $y$ on the elliptic curve \[ y^2 = (x - e_1)(x - e_2)(x - e_3), \] where $e_1$, $e_2$, and $e_3$ are distinct, is $-3$. That is, the function $y$ has a pole of order $3$ at $(0, 1, 0)$. I have no doubt that this is true; I'd like to know a simple way to see it, based on projective coordinates and independent of the fact that the sum of the orders of the zeros of $y$ is $3$ (which I understand).