# Lower bound for number of points of elliptic curve over finite field

I'm asked this kind of question:

Show there exists $$q+1$$ points on the elliptic curve over $$\mathbb{F}_q$$ given by $$y^2 = x^3-x$$ when $$q \equiv 3 (mod 4)$$

The fact that I'm asked for an approximation shows you that I don't have much tools available to prove it. Basically, I tried Hasse's theorem but I don't get reasonable conclusion (in fact I get a contradiction).

Is there a simple method to produce such an approximation?

Since $$q\equiv 3\bmod 4$$ then $$-1$$ is not a square thus for any $$a\in \Bbb{F}_q-\{0,1,-1\}$$ exactly one of $$\sqrt{a^3-a}, \sqrt{(-a)^3-(-a)}$$ is in $$\Bbb{F}_q$$ which means that
$$2\ \# \{ (a,b)\in \Bbb{F}_q{}^2, a\ne 0,1,-1,b^2=a^3-a\} = 2 (q-3)$$
Adding $$(\infty,\infty),(0,0),(1,0),(-1,0)$$ we get that $$\# \{ (a,b)\in \Bbb{F}_q{}^2, b^2=a^3-a\}\cup O = q+1$$