On the number of points on the curve $y^2 = x^{7} + ax^4 + bx$ over a finite field

We provide explicit formulae for the number of points on a genus $3$ hyperelliptic curve of type $y^2 = x^{7} + a x^{3} + b x$ over a finite field $\mathbb{F}_q$ of characteristic $p > 3$. As an application of these formulae, we prove that point-counting problem on this type of curves has heuristic time complexity of order $O(\log^4{q})$ bit operations. Tab. 2, bibliogr. 27.