Central configurations in the spatial n -body problem for $$n=5,6$$ n = 5 , 6 with equal masses

We present a computer assisted proof of the full listing of central configurations for spatial n-body problem for $$n=5$$ and 6, with equal masses. For each central configuration, we give a full list of its Euclidean symmetries. For all masses sufficiently close to the equal masses case, we give an exact count of configurations in the planar case for $$n=4,5,6,7$$ and in the spatial case for $$n=4,5,6$$ .