In The Arithmetic of elliptic Curves, on page 156, Silverman considers the map $$\phi\colon\mathbb C/\Lambda\longrightarrow E\subset \mathbb P^2(\mathbb C)$$ $$z\longmapsto [\wp(z),\wp'(z),1]$$ where $\Lambda$is a lattice, $E$ the corresponding elliptic curve $ZY^2=4X^3-g_2Z^2X-Z^3g_3$, and $g_2=g_2(\Lambda),g_3=g_3(\Lambda),\wp=\wp(\Lambda)$ as usual. The point at infinity of $E$ is $[0,1,0]$, and its preimage under the map $\phi$ should be $0$. However we cannot substitute $z=0$ into $\wp(z)$ and $\wp'(z)$ because these functions have poles at $0$. I do not understand how we get from $0\mapsto[\infty,\infty,1]$ to $0\mapsto[0,1,0]$. Must we treat this case separately? How to connect the multiplicity of poles to the order of the point at infinity?