i'm studying in German and because of corona virus we had only a video lecture, so unfortunately I have not understood how to deal in cases when I do not have the matrix. If I had it, I think I got the steps. But now I have just this table and I should determine the number of Jordan blocks. If I had the matrix it would be also pretty easy to calculate the characteristic and the minimal polynomial.
Now let $g : W \to W$ be a nilpotent endomorphism on a $12$-dimensional vector space $W$, over the field $K$, with the dimensions of $\operatorname{ker}g^i$ given by the following table: $$\begin{array}{c|c} i & 12 & 11 & 10 & 9 & 8 & 7 & 6 & 5 & 4 & 3 & 2 & 1 & 0 \\ \hline \dim \operatorname{ker} g^i & 12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 & 11 &10 & 8 & 5 & 0 \end{array}$$
I do not want the whole solution, but only a hint how to determine the matrix.
The sequence that we need to consider here is sometimes called the "Weyr characteristc" (cf. Horn and Johnson's Matrix Analysis for instance).
Since $g$ is nilpotent, its only eigenvalue is zero, and so all Jordan blocks that we refer to are associated with eigenvalue $\lambda = 0$. Let $k_i = \dim \ker g^i$ (note that $\dim \ker g^0 = \dim \ker \operatorname{id} = 0$). Let $a_i = k_{i} - k_{i-1}$. The number $a_i$ counts the number of Jordan blocks that $g$ has of size at least $i$. You should try to convince yourself that this is indeed the case.
The sequence $(a_1,a_2,a_3,\dots)$ is sometimes called the "Weyr characteristic of $g$ associated with $(g,0)$". With that established, let $b_i = a_i - a_{i-1}$. By our earlier characterization of $a_i$, we can see that $b_i$ is the number of Jordan blocks of size $i$.
So, for our example, consider the following: $$ \begin{array}{c|c} i & 0 & 1 & 2 & 3\\ \hline k_i & 0 & 5 & 8 & 10\\ \hline a_i & \cdot & 5-0=5&8-5=3 & 10-8=2\\ \hline b_i & \cdot & 5-3 = 2 & 3-2 = 1 \end{array} $$ We can continue in this fashion.