I'm trying to understand a proof in Silverman's The Arithmetic of Elliptic Curves.
Background: For an elliptic curve $E$, $x, y \in K(E)$ such that $\phi = [x, y, 1]: E \rightarrow C$ is a basepoint preserving isomorphism with a curve $C$ given by a Weierstrass equation are called Weierstrass coordinates. (By 'basepoint preserving' I mean that $\phi(O) = (0 : 1: 0)$, where $O$ is the basepoint of $E$. Put differently $\phi$ is an Isogeny if $C$ itself is viewed as an elliptic curve.)
In Silverman's proof about the relation between two pairs of Weierstrass coordinates (see The Arithmetic of Elliptic Curves, Ch. III Proposition 3.1) it is claimed that all Weierstraß coordinates $x, y$ have poles at the basepoint of orders 2 and 3, respectively.
I know that, since $\phi$ is an isomorphism, I can check this for the rational functions $X/Z$ and $Y/Z$ on $C$ instead. As $C$ is nonsingular, the line $Y = 0$ has 3 distinct intersections with C on the affine plane $Z \neq 0$, so $Y/Z$ has indeed order $-3$ at $(0 : 1 : 0)$.
But how can I proof that $X/Z$ has order $-2$ at the base point?
A similar question was asked here, but without a satisfactory answer (for me).