I am told that if $g \in A_n$ then its conjugacy class splits into two conjugacy classes of equal size if and only if the lengths in the cycle type of $g$ are distinct odd numbers.
Applying this to $A_5$, in $S_5$ we have 7 conjugacy classes with representatives
$1, (1 2), (1 2 3), (1 2 3 4), (1 2 3 4 5), (1 2)(3 4), (1 2)(3 4 5)$.
Taking the even ones we have $1, (1 2 3), (1 2 3 4 5), (1 2)(3 4)$ which have cycle types $(1), (3), (5) $ and $(2, 2)$. From what I said above it seems to say that $(1 2 3)$ and $(1 2 3 4 5)$ as they have cycles types $(3)$ and $(5)$ respectively which are odd. But this would give 6 conjugacy classes but there are only 5. What is my mistake?