  In this paper we give an algorithm to find the 3-torsion subgroup of the
Jacobian of a smooth plane quartic curve. We describe $3-$torsion points in
terms of cubics which triply intersect the curve, and use this to define a
system of equations whose solution set corresponds to the coefficients of these
cubics. We compute the points of this zero-dimensional, degree $728$ scheme
first by approximation, using homotopy continuation and Newton-Raphson, and
then using lattice reduction or continued fractions to obtain accurate
expressions for these points. We describe how the Galois structure of the field
of definition of the $3$-torsion subgroup can be used to compute local
conductor exponents, including at $p=2$.
