I have a non-hyperelliptic curve $C$ of genus 3 and I'm interested in finding the $K$-rational points on the curve with $K$ a fixed imaginary quadratic number field.

As $C$ is a non-hyperelliptic genus 3 curve, the Jacobian $J_C(\mathbb{Q})$ of $C$ cannot be computed directly in any current computer algebra system that I know of. I'm wondering if somebody would point me to the state of art so that I can figure out if $J_C(\mathbb{Q})$ can be determined.