counting points on elliptic curve

Question

Given an elliptic curve with equation $y^2=x^3+ax+b$, and i want to find the number of points $(a,b)\in E(\mathbb{F}_p)$ where the polynomial has repeated roots, how do i do it? I have an intuition it is $p$, but don't know how to rigorously show it.

Thanks

Answer 1

An elliptic curve whose polynomial has repeated roots is not in fact an elliptic curve, but a singular cubic curve, as it has a singular point where the group law breaks down. Still, we can remove those broken points and treat the remaining ones as a group. There are 3 possible cases:

• $y^2 = x^3 \bmod p$. This curve has a triple root, and is isomorphic to the additive group of integers modulo $p$. Therefore the order here is $p$.
• $y^2 = x^2(x + a) \bmod p$, square $a$. This curve has a double root, and is isomorphic to the multiplicative group of integers modulo $p$. Therefore the order here is $p-1$.
• $y^2 = x^2(x + a) \bmod p$, nonsquare $a$. This curve has a double root, and is isomorphic to the group of integers in $\mathbb{F}_{p^2}$ of norm $1$. Therefore the order here is $p + 1$.

The full treatment of all these cases is given in several places, e.g. Husemöller or Washington or Silverman.

The simplest case is the first one, where you have $y^2 = x^3$. Here's the outline: show that you can represent a point unambiguously as $t = x/y$ ($t = 0$ for the point at infinity), since $x = 1/t^2$ and $y = 1/t^3$. Since there are as many $t$ as there are field elements, the group size is the same, $p$.

The other two cases follow the similar approach of bijectively encoding a point as a single $\mathbb{F}_p^*$ (or $\mathbb{F}_{p^2}^*$) element.