Test for zero cup product

How can I check whether the cohomology ring H^*(X) of a simplicial complex X has zero cup product (like, say, for X a wedge of spheres). I have a long list of simplicial complexes, so I need a fully algorithmic approach. I probably should start with X.cohomology_ring(QQ) but I don't know what to do next... Btw, I am a sage newbie.