We study numerically the nonlinear stage of the modulational instability (MI) of cnoidal waves in the framework of the focusing one-dimensional nonlinear Schrödinger (NLS) equation. Cnoidal waves are exact periodic solutions of the NLS equation which can be represented as the lattices of overlapping solitons. The MI of these lattices leads to the development of 'integrable turbulence' (Zakharov 2009 Stud. Appl. Math. 122 219-34). We study the major characteristics of turbulence for the dn-branch of cnoidal waves and demonstrate how these characteristics depend on the degree of 'overlapping' between the solitons within the cnoidal wave. Integrable turbulence, which develops from the MI of the dn-branch of cnoidal waves, asymptotically approaches its stationary state in an oscillatory way. During this process, kinetic and potential energies oscillate around their asymptotic values. The amplitudes of these oscillations decay with time as t−α, 1<α<1.5, the phases contain nonlinear phase shift decaying as t−1/2, and the frequency of the oscillations is equal to the double maximal growth rate of the MI, s=2γmax. In the asymptotic stationary state, the ratio of potential to kinetic energy is equal to  −2. The asymptotic PDF of the wave intensity is close to the exponential distribution for cnoidal waves with strong overlapping, and is significantly non-exponential for cnoidal waves with weak overlapping of the solitons. In the latter case, the dynamics of the system reduces to two-soliton collisions, which occur at an exponentially small rate and provide an up to two-fold increase in amplitude compared with the original cnoidal wave. For all cnoidal waves of the dn-branch, the rogue waves at the time of their maximal elevation have a quasi-rational profile similar to that of the Peregrine solution.