We analyze the modes of instability of an elastic homogeneous arch that is loaded at its center. We study first the dynamical linear stability of the symmetric elastic arch with respect to small perturbations. We also perform a constrained minimization of the static energy of the system. The numerical resolution of both the dynamical and the static problems allows the determination of the phase diagram corresponding to the behavior of the arch. For this simple elastica problem, the phase diagram shows a very rich structure that is in agreement with the experimental results.