Invariants of genus 4 curves

The present paper gives an explicit classification of the isomorphism classes of non-hyperelliptic genus 4 curves over an algebraically closed field of characteristic 0. A non-hyperelliptic genus 4 curve lies on a quadric in ${\mathbb{P}}^{\mathbb{3}}$ of rank 3 or 4. In the case of rank 3, we give a set of 60 invariants which classify the isomorphism classes, and in the case of rank 4, we find 65 invariants. These invariants are defined by transvectants and can be efficiently computed on a given example.