  We consider the problem of separability of quantum channels via the Choi
matrix representation given by the Choi-Jamio{\l}kowski isomorphism. We explore
three classes of separability across different cuts between systems and
ancillae and we provide a solution based on the mapping of the coordinates of
the Choi state (in a fixed basis) to a truncated moment sequence (tms) $y$.
This results in an algorithm which gives a separability certificate using
semidefinite programming. The computational complexity and the performance of
it depend on the number of variables $n$ in the tms and on the size of the
moment matrix $M_t(y)$ of order $t$. We exploit the algorithm to numerically
investigate separability of families of 2-qubit and single-qutrit channels; in
the latter case we can provide an answer for examples explored earlier through
the criterion based on the negativity $N$, a criterion which remains
inconclusive for Choi matrices with $N=0$.
