Abstract: We study ground states of a Bose-Einstein condensate in a harmonic plus Gaussian trap. It is found that as the condensate forms a giant vortex the number of vortices is equal to the average angular momentum and the density distribution of the ground state is the same as that of angular momentum. We draw a conclusion that the ground state with giant vortex is the eigen state of angular momentum. As the potential well changes from an isotropy toroidal trap to an anisotropic toroidal trap, the ratio of the average angular momentum of the condensates to the number of vortices decreases slowly from 1, and then drops rapidly and stays near 0.5. Characteristics of density distribution of condensate and angular momentum distribution are given and explanation is shown.