  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
$\mathbb{P^3}$ 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.
