  We construct rogue wave solutions of a fifth-order nonlinear Schr\"odinger
equation on the Jacobian elliptic function background. By combining Darboux
transformation and the nonlinearization of spectral problem, we generate rogue
wave solution on two different periodic wave backgrounds. We analyze the
obtained solutions for different values of system parameter and point out
certain novel features of our results. We also compute instability growth rate
of both $dn$ and $cn$ periodic background waves for the considered system
through spectral stability problem. We show that instability growth rate
decreases (increases) for $dn$-$(cn)$ periodic waves when we vary the value of
the elliptic modulus parameter.
