Theorem: if G is a Eulerian graph of order $n$ with $k$ different Euler circuits, then line graph of $G$ has $k\cdot2^{n-1}$ different Euler circuits.
And I tried several times and I have no idea how to prove that. I can only find that line graph of Eulerian graph is a Eulerian graph and that an Euler circuit in $G$ is also a Hamiltonian cycle in $L(G)$.