Counting points on hyperelliptic curves of type $y^2=x^{2g+1} + ax^{g+1} + bx$

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 $p$.