In this paper, after investigating the modulation instability (MI) of the seventh-order nonlinear Schrödinger (NLS) equation, the rogue wave solutions on the background of the Jacobi elliptic functions dn and cn for a seventh-order NLS equation is constructed. On the background of the Jacobi elliptic function, through the approaches of the nonlinearization of spectral problem and Darboux transformation, two kinds of rogue periodic waves are obtained. Finally, the nonlinear dynamics of two kinds of rogue periodic waves are analysed.