On calculation of an automorphism group of a hyperelliptic curve

An algorithm for finding an automorphism group of a hyperelliptic curve $y^2 = p (x) $ with $p \in \mathbb Q [x] $ over field of complex numbers is proposed. The algorithm is based on parametric representation of the curve at singular points by the help of power series. The implementation of the algorithm in the computer algebra system Sage is presented, several examples are given. Numerical experiments have shown that the algorithm does not lead to excessively complex calculations. Used format for description of the found groups allows to apply standard for Sage instruments for the investigation of small-order groups.