On construction of maximal genus $3$ hyperelliptic curves

We describe two methods of contructing genus $3$ maximal hyperelliptic curves of type $y^2=x^7+ax^4+bx$ over a finite field. We consider the case when $b$ is a cubic residue in this field. In this case the Jacobian of the curve decomposes into three elliptic curves. The first method is based on finding a pair of supersingular elliptic curves over a prime field. One of the curves in the pair is chosen to have $j$-invariant equal to $0$ or $1728$. The $j$-invariant of the second elliptic curve can be computed from the $j$-invariant of the first curve using an explicit formula. After finding the pair, the maximal genus $3$ curve is constructed over a suitable extension of the finite field. This method does not allow us to enumerate all maximal curves, but gives a very efficient algorithm for the family of maximal curves. The second method is based on factorization of the Legendre polynomials, which are Hasse invariants of the elliptic curves in the Jacobian decomposition. Using this method, we construct all possible maximal hyperelliptic curves over $\mathbb{F}_{p^2}$ for $a \neq 0, b = 1$ and $p \leq 7151$.