Point count for y squared equals x cubed minus x minus one over finite extensions of F3
= Point count for y squared equals x cubed minus x minus one over finite extensions of F3
{c}
{title2=$\#E(\mathbb F_{3^r})$ for $E:y^2=x^3-x-1$}
For $E:y^2=x^3-x-1$ over $\mathbb F_3$, the <Trace of Frobenius> is three and
$$
\#E(\mathbb F_{3^r})
=3^r+1-2\cdot3^{r/2}\cos(r\pi/6).
$$
In particular, $\#E(\mathbb F_{3^r})=3^r+1$ exactly when $r\equiv3\pmod6$.