Here are two illustrative examples of computations described in [F.-O. Schreyer, M. Weimann, J. Schicho, Computational aspects of gonal maps and radical parametrization of curves], Example 4.12 and Example 4.13, in the computer algebra system Magma.
The input files are magmain1 and magmain2. To produce the results magmaout1 and magmaout2, run the command magma < magmain1 or magma < magmain2 on a computer with Magma installed. Alternatively, copy the file into the input window of the Magma Online Calculator.