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

We present a computer assisted proof of the full listing of central configurations for spatial n-body problem for $$n=5$$ 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 $$n=4,5,6,7$$ n = 4 , 5 , 6 , 7 and in the spatial case for $$n=4,5,6$$ n = 4 , 5 , 6 .