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

In this work, we investigate hyperelliptic curves of type $C: y^2 = x^{2g+1}

• ax^{g+1} + bx$overthefinitefield$$\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 $p$.