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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?

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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 $$\# \{ (a,b)\in \Bbb{F}_q{}^2, b^2=a^3-a\}\cup O = q+1$$

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