Thermodynamic and mechanical properties of the six known phases in the La–Mg phase diagram, viz. LaMg, LaMg₂, LaMg₃, La₅Mg₄₁, La₂Mg₁₇, and LaMg₁₂, and their elemental antecedents, Mg and La, are computed with density functional theory (DFT) using the PBE and PBEsol exchange-correlation functionals. Phase stability analyses show that both LaMg₂ and La₅Mg₄₁ are metastable at low temperatures which is consistent with experiments and vibrational spectra. We generalize an existing approach for computing the crystallographic dependence of Young's modulus and Poisson's ratio, which is presently limited to cubic systems, to address any space group symmetry using 0 K elasticity tensor components (Cij) from DFT. Isothermal and isentropic Cij(T) are computed with the quasiharmonic approximation (QHA) as are the linear thermal expansion of the cubic compounds, the average linear thermal expansion for the non-cubic compounds, the bulk modulus, and the constant pressure heat capacity. A critical comparison of theoretical results from the PBE and PBEsol functionals is made with available experimental data.