Number of Points on an Elliptic Curve

Question

If I have an elliptic curve

$$E: y^2 = x^3 + bx + c$$, with $b, c$ integers mod some prime $p$.

And $x^3 + bx + c$ has at least one root mod $p$.

How can I show that the number of points on the curve is even?

Answer 1

The number of points is $1+\sum_{x\in \mathbb F_p}\left(\left(\frac{x^3+ax+b}{p}\right)+1\right)$, where $\left(\frac{x^3+ax+b}{p}\right)$ is the Legendre symbol modulo $p$. If $x$ is not a root of $x^3+ax+b$, the term inside the sum is either $0$ or $2$. Otherwise, it is $1$. If $x^3+ax+b$ has a root, then it has either $1$ or $3$ roots and the sum is odd, which makes the number of points even.

More conceptually, if that polynomial has a root than $E$ has a rational $2$-torsion points, and therefore $2$ divides the order of $E(\mathbb F_p)$.