How can a renormalization group fixed point be scale invariant without being conformal? Polchinski (1988) showed that this may happen if the theory contains a virial current -- a non-conserved vector operator of dimension exactly (d−1), whose divergence expresses the trace of the stress tensor. We point out that this scenario can be probed via lattice Monte Carlo simulations, using the critical 3d Ising model as an example. We measure the scaling dimension of the lowest virial current candidate V, and find \Delta_V=6\pm1, well above 2 expected for the true virial current. Our result shows that the critical 3d Ising model has no virial current, providing a structural explanation for the conformal invariance of the model.