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In algebraic geometry, it is important to provide effective parametrizationsfor families of curves, both in theory and in practice. In this paper, wepresent such an effective parametrization for the moduli of genus-$5$ curvesthat are neither hyperelliptic nor trigonal. Subsequently, we construct analgorithm for a complete enumeration of non-special genus-$5$ curves havingmore rational points than a specified bound, where ``non-special curve'' meansthat the curve is non-hyperelliptic and non-trigonal with mild singularities ofthe associated sextic model that we propose. As a practical application, weimplement this algorithm using the computer algebra system MAGMA, specificallyfor curves over the prime field of characteristic $3$.

Representation of non-special curves of genus 5 as plane sextic curves and its application to finding curves with many rational points