Let $n\geq 1$ be an integer and $b \in \mathbb{F}_{3^{n}}$ be an element such that $b^{\frac{3^{n}-1}{2}}=(-1)^{n+1}$ (we fix $n$ and $b$).
Let $C$ be the curve defined over $\mathbb{F}_{3^{n}}$ by the equation $$y^{3^n+1}-y^2 = x^3 + bx.$$ I would like to compute the number of $\mathbb{F}_{3^{2nj}}$ -rational points of $C$ for $j \geq 1$ any integer.
Is there a good way to do this? I would appreciate any suggestions or information.
( I have already computed the case $j=1$ : I get $$\#C(\mathbb{F}_{3^{2n}})=1+3^{n+1}+3^n(3^n-1)$$ by explicit computations of the equation above (the first "$1$" is for the point at infinity). However, the same computation is very complicated when $j$ is arbitrary...)