I'll use a longer cycle to help describe two techniques for writing disjoint cycles as the product of transpositions:
Let's say $\tau = (1, 3, 4, 6, 7, 9) \in S_9$
Then, note the patterns:
Method 1: $\tau = (1, 3, 4, 6, 7, 9) = (1, 9)(1, 7)(1, 6)(1, 4)(1, 3)$
Method 2: $\tau = (1, 3, 4, 6, 7, 9) = (1, 3)(3, 4)(4, 6)(6, 7)(7, 9)$
Both products of transpositions, method $1$ or method $2$, represent the same permutation, $\tau$. Note that the order of the disjoint cycle $\tau$ is $6$, but in both expressions of $\tau$ as the product of transpositions, $\tau$ has $5$ (odd number of) transpositions. Hence $\tau$ is an odd permutation.
Now, don't forget to multiply the transpositions you obtain for each disjoint cycle so you obtain an expression of the permutation $S_{11}$ as the product of the product of transpositions, and determine whether it is odd or even:
$\sigma = (1, 4, 10)(3, 9, 8, 7, 11)(5, 6)$.
Best Answer
Once a permutation is known as a product of disjoint (hence commuting) cycles, one can work in parallel and so it suffices to know how to write a cycle as a product of transpositions. To avoid a clutter of subscripts, I'll give an example:
$(7413625) = (75)(72)(76)(73)(71)(74)$
(In each of these transpositions 7 occurs, the other numbers accompanying 7 are taken in the reverse order to the given cycle)