Point count for y squared equals x cubed minus x minus one over finite extensions of F3 ($\#E({\mathbb{F}}_{3^r})$ 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 andIn particular, $\#E({\mathbb{F}}_{3^r})=3^r+1$ exactly when $r\equiv 3(\mathrm{mod}6)$.