I am trying to calculate the cardinality of a genus $g=2$ curve under certain hypotheses, but I do not know if the information I have is enough to infer this number or a bound for it (without using the Hasse inequality or other fancy bounds).
Let $H/\mathbb{F}_q$ be a hyperelliptic curve of genus $2$ and $J$ its Jacobian. Suppose that there is an $n\in\mathbb{Z}$ such that for $\phi,[n]\in\text{End}(J)$ we have that $\phi+[n]=0$ where $\phi$ is the $q$-Frobenius Endomorphism.
What I want is to calculate the number of points of $\#H(\mathbb{F}_q)=?$ directly without using a fancy inequality (like Hasse-Weil).
I have tried the following:
Since $\phi=-[n]\in\text{End}(J)$ we have that $\deg\phi=\deg [n]$, therefore $q^g = n^{2g}$ which in our case implies that $q^2 = n^4$. Therefore $q=n^2=(p^k)^2$ is a square.
Also we have that $\phi$ satisfies its characteristic polynomial $\chi_\phi(X)$ on the Tate module and since $\phi=[-n]=[p^k]$ we have that $\chi_\phi(X)=(X+p^k)^4$.
This $\textit{smells}$ to a supersingular curve $H$. I think that using the previous, I can infer that $J[q]=\{0\}$ (please correct me). And if this is true then $q|\#H(\mathbb{F}_q)-1$.
Also an easy counting argument tells me that $\#J(\mathbb{F}_q)=\frac{\#H(\mathbb{F}_q)^2+\#H(\mathbb{F}_{q^2})}{2}-q$ which can be found in Cassels and Flynn page 80, therefore: $\#J(\mathbb{F}_q)=\chi_\phi(1)=(1+p^k)^4=\frac{(\#H(\mathbb{F}_q)^2+\#H(\mathbb{F}_{q^2})}{2}-q$.
I do not know where to go next, Is it possible to get $\#H(\mathbb{F}_q)$ with this information? or at least a bound ?