Suppose we have a matrix $A \in M_{12}(\mathbb{C})$ whose Jordan canonical form is
$$J_1(0) \oplus J_1(0) \oplus J_2(0) \oplus J_2(0) \oplus J_1(\sqrt{2}) \oplus J_1(\sqrt{2}) \oplus J_1(-\sqrt{2}) J_1(-\sqrt{2}) \oplus J_1(i\sqrt{2}) \oplus J_1(-i\sqrt{2})$$
Then how would we go about finding the invariant factors given this? And how do we tell how many invariant factors there are going to be?
I know that the first is going to the minimal polynomial which must be
$$m_A(t) = t^2(t^2-2)(t^2+2)$$
and they have to product together to give the characteristic polynomial, which is
$$p_A(t)= t^6(t^2-2)^2(t^2+2)$$
Thanks in advance!