On pages 205-206 of Algebraic Curves and Riemann Surfaces, Miranda uses the canonical map of an algebraic curve to classify non-hyperelliptic curves of genus 3 as smooth plane quartics defined by the vanishing of a quartic polynomial. At some point, Miranda argues that no polynomial of degree less than four can vanish on such a curve for "degree reasons".
What degree reasons are these? I understand that we can't have degrees 1 and 2 since then the curve would have genus 0. I also understand that we can't have degree 3 since then the curve would have genus 1. What I don't understand is why we can rule out these cases for "degree reasons".