  We use invasion percolation to compute highly-accurate numerical values for
bond and site percolation thresholds p_c on the hypercubic lattice Z^d for d =
4,,,,,13. We also compute the Fisher exponent tau governing the cluster size
distribution at criticality. Our results support the claim that the mean-field
value tau = 5/2 holds for d >= 6, with logarithmic corrections to power-law
scaling at d=6.
