I have a function y^2 = x^3 + Ax + B mod p. I know the curve has a singularity as the discriminate is zero mod p. I'm trying to isolate the non-singular points of the curve by mapping the singularity to infinity. I've read a lot about the steps I'm supposed to follow - find the singularity S, calculate the distinct tangents at S (y = a1x + b1 and y = a2x + b2) and then apply the mapping, but I can't find any practical examples on how to actually do this.
Can anyone please point me in the direction of a similar calculation using actual numbers, instead of a proof?
Thanks very much!
I think that what you want is described in Proposition 2.5 of Silverman's Arith. of Elliptic Curves: If your elliptic curve $E$ has distinct tangent lines at the singularity, then $E_{\rm ns} \to \bar K^*$ given by $(x,y) \mapsto \frac{y-a_1x-b_1}{ y-a_2x-b_2}$ is an isomorphism of groups. If $E$ has a cusp (the tangent lines are coincide), then $E_{\rm ns}\to \bar K^{+}$ given by $(x,y) \mapsto \frac{x-x(S)}{y-a x-b}$ is again an isomorphism (where $S$ denotes de singular point, while $x(S)$ is the coordinate of $S$)
ADDED: If you really want to work with numbers why don't you construct some of them? Take the simplest example of singular cubic curve you know, i like $C:Y^2 =X^3$. Find the tangent lines, that is, only $Y=0$ since $(0,0)$ is a cusp. Now observe that the map I described above is given by $(0,0)\neq(x,y) \mapsto x/y$, which gives you the bijection between $C$ and $K$ 'sending' $(0,0)$ to infinty.