# possible Jordan normal forms of endomorphism

Let $$f:\mathbb{C}^6\to\mathbb{C}^6$$ be a linear map with characteristic polynomial $$ch_f(X)=(X+2)^4(X-1)^2$$ and $$rank(f+2id)>rank(f+2id)^2=1$$, as well as $$rank(f-id)=5$$.

Assignment: Find the minimal polynomial and all possible Jordan canonical forms of $$f$$ (Hint: There are $$2$$ different Jordan canonical forms).

I can see that we have exactly one Jordan Block for the eigenvalue $$1$$ but, given the above information, it seems to me that the only other things we can conclude are that we have at most $$3$$ Jordan blocks for the eigenvalue $$-2$$, with at least one Jordan Block of size $$\geq2$$ (by elementary canonical form theory) and that gives us a total of $$4$$ possible Jordan canonical forms, as opposed to the hint above.

Thank you very much in advance.