We investigate numerically the vibrational behavior of a simple finite element model that stands for an elastic container filled with an inviscid fluid. The underlying mathematical model is detailed and its spectra is characterized. The finite element method relies upon the added-mass formulation of Morand and Ohayon. A parametric study allows to characterize the system's response to dimensionless parameters in terms of eigenfrequencies. Then an insight into the mode shapes is provided, with a discussion on the presence and the behavior of singular modes caused by specific boundary conditions in the fluid.