  We study the problem of generating the endomorphism ring of a supersingular
elliptic curve by two cycles in $\ell$-isogeny graphs. We prove a necessary and
sufficient condition for the two endomorphisms corresponding to two cycles to
be linearly independent, expanding on the work in Kohel's thesis. We also give
a criterion under which the order generated by two cycles is not a maximal
order. We give some examples in which we compute cycles which generate the full
endomorphism ring. The most difficult part of these computations is the
calculation of the trace of these cycles. We show that a generalization of
Schoof's algorithm can accomplish this computation efficiently.
