Construct (representatives of) the similarity classes of matrices over having minimal polynomial .

Recall that the minimal polynomial of a matrix is its divisibility-largest invariant factor (Prop. 13 in D&F), that the characteristic polynomial of a matrix is the product of its invariant factors (Prop. 20 in D&F) and that the degree of the characteristic polynomial of a matrix is the dimension of the matrix (By the definition of the characteristic polynomial). To construct the rational canonical forms of matrices having minimal polynomial , it suffices to construct all the possible lists of invariant factors which terminate in and whose product has degree 6.

Note that has five nontrivial divisors over : , , , , and . These are the possible next-to-last entries in a list of invariant factors which terminates in . Note that for each of these divisors, the remaining invariant factors are determined by degree considerations except in the case , where we have two possible choices for the third invariant factor.

The lists of invariant factors are as follows.

- , , ,
- , ,
- , , ,
- , ,
- , ,
- ,

The corresponding similarity class representatives (i.e. rational canonical forms) are as follows.

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