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

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 $$\mathbb{F}_p$$.

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.