In this work, we investigate hyperelliptic curves of type $C: y^2 = x^{ 2g+1} + ax^{ g+1} + bx$ over the finite field $\mathbb{ F}_q, q = p^n, p > 2$. For the case of $g = 3$ and $4$ we propose algorithms to compute the number of points on the Jacobian of the curve with complexity $\tilde{ O}(\log^4{ p})$ and $\tilde{ O}(\log^8{ p})$. For curves of genus $2-7$ we give a complete list of the characteristic polynomials of Frobenius endomorphism modulo $p$.
Categories: math.NT