reasoning through this answer is probably best... the equal sides must be long enough that the third side "fits", so this means that a triangle such as "1,1,11" would not work because the two 1-inch side could in no way touch the ends of the 11 inch side

the sum of the equal sides have to be larger than the third side (think of this in the extreme, where if the equal sides were laid out end-to-end and they were equal to the third side, then there is no triangle at all, simply two parallel lines)

so we require that the sum of the equal sides be greater than the third side and we are looking for integer solutions:

13/2 = 6.5, so our equal sides have to sum to something greater than 6.5 and therefore 4 is the smallest choice we can make, therefore we have these solutions:

4,4,(13-8) = 4, 4, 5

5,5,(13-10) = 5, 5, 3

6,6,(13-12) = 6, 6, 1

there are three choices