  We prove that there are >>X^{1/30}/(log X) imaginary quadratic number fields
with an ideal class group of 3-rank at least 5 and discriminant bounded in
absolute value by X. This improves on an earlier result of Craig, who proved
the infinitude of imaginary quadratic fields with an ideal class group of
3-rank at least 4. The proofs rely on constructions of Mestre for j-invariant 0
elliptic curves of large Mordell-Weil rank, and a method of the first author
and Gillibert for constructing torsion in ideal class groups of number fields
from rational torsion in Jacobians of curves. We also consider analogous
questions concerning rational 3-torsion in hyperelliptic Jacobians.
