The problem of thermal convection between rotating rigid plates under the influence of gravity is treated numerically. The approach uses solenoidal basis functions and their duals which are divergence free. The representation in terms of the solenoidal bases provides ease in the implementation by a reduction in the number of dependent variables and equations. A Galerkin procedure onto the dual solenoidal bases is utilized in order to reduce the governing system of partial differential equations to a system of ordinary differential equations for subsequent parametric study. The Galerkin procedure results in the elimination of the pressure and is facilitated by the use of Fourier-Legendre spectral representation. Numerical experiments on the linear stability of rotating thermal convection and nonlinear simulations are performed and satisfactorily compared with the literature.