The author proves bounds on the rank of an elliptic curve over an arbitrary number field \(k\) containing the cube roots of unity, in case the curve has a rational point of order 3. The bound involves the degree of \(k\), and the difference between the class group of \(k\) and part of the class group of the field \(K\) obtained by adjoining the coordinates of all points of order 3 to \(k\). More precisely, that field is a Kummer extension of \(k\) and one considers the ideal classes which are fixed by the action of the Galois group of \(K\) over \(k\) (so-called ambiguous classes). As a special case, we mention that if the \(j\)-invariant of the curve is nonzero and all 3-torsion points are rational over \(k\), then the bound obtained equals the degree \([k:\mathbb{Q} ]\). Standard methods of `descent by 3-isogeny' are used to relate the rank to the size of certain Galois cohomology groups. The proof is then completed by using results of Federer connecting this with ambiguous ideal classes.