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 and
$$
\#E(\mathbb F_3)=1,
\qquad a_3=3+1-1=3.
$$
The eigenvalues of the <Frobenius isogeny> are the roots
$$
\alpha,\beta=\frac{3\pm i\sqrt3}{2}=\sqrt3e^{\pm i\pi/6}
$$
of $T^2-3T+3$. The <elliptic-curve point count over a finite field> gives
$$
\#E(\mathbb F_{3^r})=3^r+1-\alpha^r-\beta^r
=3^r+1-2\cdot3^{r/2}\cos(r\pi/6).
$$
The last term vanishes exactly when $r\pi/6\equiv\pi/2\pmod\pi$. Hence
$$
\boxed{\#E(\mathbb F_{3^r})=3^r+1\quad\Longleftrightarrow\quad r\equiv3\pmod6.}
$$

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-125/1/b/solution