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In this paper we give an algorithm to find the 3-torsion subgroup of theJacobian of a smooth plane quartic curve with a marked rational point. Wedescribe $3-$torsion points in terms of cubics which triply intersect thecurve, and use this to define a system of equations whose solution setcorresponds to the coefficients of these cubics. We compute the points of thiszero-dimensional, degree $728$ scheme first by approximation, using homotopycontinuation and Newton-Raphson, and then using continued fractions to obtainaccurate expressions for these points. We describe how the Galois structure ofthe field of definition of the $3$-torsion subgroup can be used to computelocal wild conductor exponents, including at $p=2$.

Three-Torsion Subgroups and Wild Conductor Exponents Plane Quartics