  The entanglement entropy of the incompressible states of a realistic quantum
Hall system in the second Landau level are studied by direct diagonalization.
The subdominant term to the area law, the topological entanglement entropy,
which is believed to carry information about topologic order in the ground
state, was extracted for filling factors nu = 12/5 and nu = 7/3. While it is
difficult to make strong conclusions about nu = 12/5, the nu = 7/3 state
appears to be very consistent with the topological entanglement entropy for the
k=4 Read-Rezayi state. The effect of finite thickness corrections to the
Coulomb potential used in the direct diagonalization are also systematically
studied.
