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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.

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    $\begingroup$ Writing the equation of the curve would help a lot. If it has e.g. CM then there would be more tricks available. $\endgroup$
    – Daniel Loughran
    Commented Dec 27, 2022 at 9:51
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    $\begingroup$ The paper Generalized explicit descent and its applications to genus 3 curves, published in Forum of Math. Sigma in 2016, by Nils Bruin, Bjorn Poonen, and Michael Stoll gives a procedure for doing descent on the Jacobian of a genus 3 curve. $\endgroup$
    – Jeremy Rouse
    Commented Dec 27, 2022 at 15:02

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