I am looking to find the intersection number of the affine plane curves $F=Y^2-X^3+X$ and $G=(X^2+Y^2)^3-4X^2Y^2$, and I need to do it with the order function of the local ring of $F$ at the origin.
So my attempt goes as follows:
$$k[X,Y] \twoheadrightarrow \Gamma(F)=k[X,Y]/(F) \hookrightarrow \mathcal{O}_{(0,0)}(F)$$ $$ G \mapsto \bar{G} = G\mod (F) \mapsto \frac{\bar{G}}{1} $$
Then I get the intersection number as $I((0,0),F\cap G) = \text{Ord}_{(0,0)}^F(\bar{G})$.
For the map onto the coordinate ring of $F$, I wrote $X\mapsto x, Y\mapsto y : y^2=x^3-x$. From there I thought that $\bar{G} = (x^2+(x^3-x))^3 - 4x^2(x^3-x)$. The map into the local ring (which I know should be a DVR as the origin is a simple point of $F$) is a natural injection, so I thought I should be able to write $\bar{G} = u\cdot t^n$ as one can do in a DVR. I can't! I'm not sure what is going wrong.
Hint. The uniformizing parameter of $\mathcal O=k[X,Y]_{(X,Y)}/(F)$ is $y$ (the residue class of $Y$ modulo $(F)$).