I would like to work through computing the blow-up of a particular curve along a subvariety consisting of just two points, both of which are *ordinary double points.*

Let $$F(X,Y,Z) = X^4 + Y^4 - X Y^2 Z - X^2 Z^2 + Y^2 Z^2 + Z^4 \in \mathbb{F}_3[X,Y,Z]$$ and let $C \subseteq \mathbb{P}^2 $ be the curve over $\mathbb{F}_3$ defined by $F(X,Y,Z) = 0$.

Now consider the subvariety of $C$ defined by $X^2 + Z^2 = Y = 0$, i.e. the two points $[2\alpha + 2, 0, 1], [\alpha + 1, 0, 1]$ where here $\alpha$ satisfies $\alpha^{2} + 2\alpha +2 = 0$.

I would like to compute the blow-up of $C$ along this subvariety. I know how to compute blow-ups at a single point but I am not sure how to go about computing this blow-up along *two* points.

EDIT: I forgot to emphasise that I am interested in blowing up a curve along a subvariety of just two points, where those points are (conjugate) *ordinary double points*.