  In algebraic geometry, it is important to give good parametrizations of
spaces of curves, theoretically and also practically. In particular, the case
of non-hyperelliptic curves is the central issue. In this paper, we give a very
effective parametrization of curves of genus $5$ which are neither
hyperelliptic nor trigonal. After that, we construct an algorithm for a
complete enumeration of generic curves of genus $5$ with many rational points,
where "generic" here means non-hyperelliptic and non-trigonal with mild
singularities of the associated sextic model which we propose. As an
application, we execute an implementation on computer algebra system MAGMA of
the algorithm for curves over the prime field of characteristic $3$.
