According to this lecture, how can a hyperelliptic curve (of genus $2$) be constructed in the following example?
Let $C: f(x)=y^2$ (where $f(x)$ is of degree $5$) be the curve we want to construct over the finite field $\mathbb F_{59}$.
Suppose the characteristic polynomial $P$ of $C$ is given by $x^4 + 3x^3 + 99x^2 + 177x + 3481$. Let $J(C)$ denote the Jacobian group of $C$.
From this article, We know that $|J|$ is either $P(1) = 3761$ or $P(-1) = 3401$. Similar to elliptic curves, $C$ should have Complex Multiplication or CM.
With elliptic curves, one needs to computer the Hilbert Class polynomial $H$ of the characteristic polynomial (a quadratic), then find an elliptic curve whose $j$-invariant is equal a root of $H$.
In the hyperelliptic case, the number field obtained by the characteristic polynomial $P$ has class number $4$, so the Hilbert class polynomial $H$ of $P$ should also be of degree $4$.
The first problem I run into is computing $H$. Secondly, assuming we know $H$, is there a function similar to the $j$-invariant that can be used to calculate the curve? The first lecture mentions Igusa invariants, yet there is no explicit formula to compute such an invariant of a hyperelliptic curve (analogous to $j$-invariants of elliptic curves).
In summary, the Jacobian of our curve $C$ over finite field $F_{59}$ must have cardinality $3401$ or $3761$, and the characteristic polynomial of $C$ is $P$. The problem is constructing $C$ as mentioned above.
Any suggestions? Thanks for help in advance.