  Lattice models forming bands with higher Chern number offer an intriguing
possibility for new phases of matter with no analogue in continuum Landau
levels. Here, we establish the existence of a number of new bulk insulating
states at fractional filling in flat bands with Chern number $C=N>1$, forming
in a recently proposed pyrochlore model with strong spin-orbit coupling. In
particular, we find compelling evidence for a series of stable states at
$\nu=1/(2N+1)$ for fermions as well as bosonic states at $\nu=1/(N+1)$. By
examining the topological ground state degeneracies and the excitation
structure as well as the entanglement spectrum, we conclude that these states
are Abelian. We also explicitly demonstrate that these states are nevertheless
qualitatively different from conventional quantum Hall (multilayer) states due
to the novel properties of the underlying band structure.
