Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 125 / 1 / b / Solution

For $x=0,1,2$, the value of $x^3-x-1$ is the nonsquare $2$ in ${\mathbb{F}}_3$. Thus the point at infinity is the only rational point andThe eigenvalues of the Frobenius isogeny are the rootsof $T^2-3T+3$. The elliptic-curve point count over a finite field givesThe last term vanishes exactly when $r\pi /6\equiv \pi /2(\mathrm{mod}\pi )$. Hence