We report in this paper the first principle calculation of elastic constants of metal aluminium using a model of cluster Al₁₃ with face-centered-cubic structure. Total energies of Al₁₃ are first calculated at various lattice spacings using the M_s-X_a method，after that total energies are fitted using morse potential function，the porameters of which are finally used to calculate the elastic constants of aluminium.The results are: C₁₁=8.83×l0¹¹ dyn/cm² and C₁₂=6.73×10¹¹dyn/cm²，which are in good agreement with ather theories and experiment.