Let $k$ be an algebraic closed field with character not equal to 2, $a,b,c\in k$ be distinct numbers, and consider the curve $C: y^2=(x-a)(x-b)(x-c)$. Let $P=(a,0),P_{\infty}$ for the point at infinity. Then $div(x-a)=2(P)-2(P_{\infty})$. Here, the curve $C$ really means projective curve $C: y^2z=(x-az)(x-bz)(x-cz)$, so I guess $div(x-a)$ here should mean $div(\frac{x}{z}-a)$.
My problem is how to compute $ord_{P_{\infty}}(\frac{x}{z}-a)$. I konw the local ring at $P_{\infty}$ is $(k[x,z]/(z-(x-az)(x-bz)(x-cz)))_{(x,z)}$, and it is a regular local ring. But I cannot find the uniformizer and prove $ord_{P_{\infty}}(\frac{x}{z}-a)=2$.