The sum of numbers of points of two non-isomorphic elliptic curves over a finite field

Question

Suppose that $p\geq 3$ is a prime and let $\alpha\in \mathbb{F}_p$ be a non-square. Let $E$ be an elliptic curve $y^2=x^3+ax^2+bx+c$ over $\mathbb{F}_p$ and let $E'$ be the elliptic curve $\alpha y^2=x^3+ax^2+bx+c$ over $\mathbb{F}_p$. Show that the equality $2p+2=\#E(\mathbb{F}_p)+\#E'(\mathbb{F}_p)$ where $\#E(\mathbb{F}_p)$ denote the number of points of $E$ over $\mathbb{F}_p$.

Answer 1

For every $x$, either $x^3+ax^2+bx+c$ is a square or it is not. Don't forget to add the point at infinity.