Summary: Hyperelliptic curves have been widely studied for cryptographic applications, and some special hyperelliptic curves are often considered to be used in practical cryptosystems. Computing Jacobian group orders is an important operation in constructing hyperelliptic curve cryptosystems, and the most common method used for the computation of Jacobian group orders is by computing the zeta functions or the characteristic polynomials of the related hyperelliptic curves. For the hyperelliptic curve \(C_q:v^2 = u^p + au + b\) over the field \(\mathbb{F}_q\) with \(q\) being a power of an odd prime \(p\), Duursma and Sakurai obtained its characteristic polynomial for \(q = p\), \(a = -1\), and \(b \in \mathbb{F}_p\). In this paper, we determine the characteristic polynomials of \(C_q\) over the finite field \(\mathbb{F}_{p^n}\) for \(n = 1, 2\) and \(a, b \in \mathbb{F}_{p^n}\). We also give some computational data which show that many of those curves have large prime factors in their Jacobian group orders, which are both practical and vital for the constructions of efficient and secure hyperelliptic curve cryptosystems.