Number of points on an elliptic curve over $ \mathbb{F}_{q} $.

Question

I have the following elliptic curve: $$ E: \quad Y^{2} = X^{3} + 1 ~ \text{over} ~ \mathbb{F}_{q}, ~ \text{where} ~ q \equiv 1 ~ (\text{mod} ~ 3). $$ I want to know the number of points on this curve. Any ideas on how I could approach the problem?

Answer 1

When $q \equiv 1 \pmod 3$ and $q>3$ , there is a unique way (up to the sign of $b$) to write $q = a^2+3b^2$ with $a \equiv 1 \pmod 3$.

I think the explicit class field theory for $\Bbb Q(\sqrt{-3})$ can make a parallel between the action of $Frob_q$ on the points of finite order of the elliptic curve and the multiplication by $a+b\sqrt{-3}$ (or its conjugate) on $(\Bbb Q/\Bbb Z)^2$. Then the number of points is the number of fixpoints of $Frob_q$ which is $det(Frob_q-1) = Norm(a-1+b\sqrt{-3}) = q+1-2a$.