4

${\mathcal{A}}_2$ or

${\mathcal{A}}_2$

$\begin{array}{l}\left\{\begin{array}{cc}1& \begin{array}{cc}8& \begin{array}{cc}27& \begin{array}{cc}64& \begin{array}{cc}125& \cdots \end{array}\end{array}\end{array}\end{array}\end{array}\right\}\\ \oplus \left\{\begin{array}{cccccc}\left(\begin{array}{c}-2\\ 0\end{array}\right)& \left(\begin{array}{c}-3\\ 0\end{array}\right)& \left(\begin{array}{c}-4\\ 0\end{array}\right)& \left(\begin{array}{c}-4\\ 0\end{array}\right)& \left(\begin{array}{c}-4\\ 0\end{array}\right)& \\ & \left(\begin{array}{c}-3\\ 1\end{array}\right)& \left(\begin{array}{c}-4\\ 1\end{array}\right)& \left(\begin{array}{c}-4\\ 1\end{array}\right)& \left(\begin{array}{c}-4\\ 1\end{array}\right)& \\ & & \left(\begin{array}{c}-4\\ 2\end{array}\right)& \left(\begin{array}{c}-4\\ 2\end{array}\right)& \left(\begin{array}{c}-4\\ 2\end{array}\right)& \cdots \\ & & & \left(\begin{array}{c}-4\\ 3\end{array}\right)& \left(\begin{array}{c}-4\\ 3\end{array}\right)& \\ & & & & \left(\begin{array}{c}-4\\ 4\end{array}\right)& \end{array}\right\}=\left\{\begin{array}{cc}1& \begin{array}{cc}5& 1\end{array}\end{array}\right\}\end{array}$

5

${\mathcal{A}}_3$ or

${\mathcal{A}}_3$

$\begin{array}{l}\left\{\begin{array}{cc}1& \begin{array}{cc}8& \begin{array}{cc}27& \begin{array}{cc}64& \begin{array}{cc}125& \cdots \end{array}\end{array}\end{array}\end{array}\end{array}\right\}\\ \oplus \left\{\begin{array}{cccccc}\left(\begin{array}{c}-3\\ 0\end{array}\right)& \left(\begin{array}{c}-4\\ 0\end{array}\right)& \left(\begin{array}{c}-4\\ 0\end{array}\right)& \left(\begin{array}{c}-4\\ 0\end{array}\right)& \left(\begin{array}{c}-4\\ 0\end{array}\right)& \\ & \left(\begin{array}{c}-4\\ 1\end{array}\right)& \left(\begin{array}{c}-4\\ 1\end{array}\right)& \left(\begin{array}{c}-4\\ 1\end{array}\right)& \left(\begin{array}{c}-4\\ 1\end{array}\right)& \\ & & \left(\begin{array}{c}-4\\ 2\end{array}\right)& \left(\begin{array}{c}-4\\ 2\end{array}\right)& \left(\begin{array}{c}-4\\ 2\end{array}\right)& \cdots \\ & & & \left(\begin{array}{c}-4\\ 3\end{array}\right)& \left(\begin{array}{c}-4\\ 3\end{array}\right)& \\ & & & \left(\begin{array}{c}-4\\ 4\end{array}\right)& \left(\begin{array}{c}-4\\ 4\end{array}\right)& \end{array}\right\}=\left\{\begin{array}{cc}1& \begin{array}{cc}4& 1\end{array}\end{array}\right\}\end{array}$

6

$Ker$

$\begin{array}{l}\left\{\begin{array}{cc}1& \begin{array}{cc}8& \begin{array}{cc}27& \begin{array}{cc}64& \begin{array}{cc}125& \begin{array}{cc}216& \begin{array}{cc}343& \cdots \end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\right\}\\ \oplus \left\{\begin{array}{cccccc}\left(-4\right)& \left(\begin{array}{c}-4\\ 0\end{array}\right)& \left(\begin{array}{c}-4\\ 0\end{array}\right)& \left(\begin{array}{c}-4\\ 0\end{array}\right)& \left(\begin{array}{c}-4\\ 0\end{array}\right)& \\ & \left(\begin{array}{c}-4\\ 1\end{array}\right)& \left(\begin{array}{c}-4\\ 1\end{array}\right)& \left(\begin{array}{c}-4\\ 1\end{array}\right)& \left(\begin{array}{c}-4\\ 1\end{array}\right)& \\ & & \left(\begin{array}{c}-4\\ 2\end{array}\right)& \left(\begin{array}{c}-4\\ 2\end{array}\right)& \left(\begin{array}{c}-4\\ 2\end{array}\right)& \cdots \\ & & & \left(\begin{array}{c}-4\\ 3\end{array}\right)& \left(\begin{array}{c}-4\\ 3\end{array}\right)& \\ & & & & \left(\begin{array}{c}-4\\ 4\end{array}\right)& \end{array}\right\}=\langle \begin{array}{cc}1& \begin{array}{cc}4& 1\end{array}\end{array}\rangle \end{array}$

7

${\mathcal{A}}_4$

or

${\mathcal{A}}_4$

$\begin{array}{l}\left\{\begin{array}{cc}1& \begin{array}{cc}8& \begin{array}{cc}27& \begin{array}{cc}64& \begin{array}{cc}125& \begin{array}{cc}216& \begin{array}{cc}343& \cdots \end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\right\}\\ \oplus \left\{\begin{array}{cccccc}\left(-5\right)& \left(\begin{array}{c}-5\\ 0\end{array}\right)& \left(\begin{array}{c}-5\\ 0\end{array}\right)& \left(\begin{array}{c}-5\\ 0\end{array}\right)& \left(\begin{array}{c}-5\\ 0\end{array}\right)& \left(\begin{array}{c}-5\\ 0\end{array}\right)\\ & \left(\begin{array}{c}-5\\ 1\end{array}\right)& \left(\begin{array}{c}-5\\ 1\end{array}\right)& \left(\begin{array}{c}-5\\ 1\end{array}\right)& \left(\begin{array}{c}-5\\ 1\end{array}\right)& \left(\begin{array}{c}-5\\ 1\end{array}\right)\\ & & \left(\begin{array}{c}-4\\ 2\end{array}\right)& \left(\begin{array}{c}-5\\ 2\end{array}\right)& \left(\begin{array}{c}-5\\ 2\end{array}\right)& \left(\begin{array}{c}-5\\ 2\end{array}\right)\\ & & & \left(\begin{array}{c}-5\\ 3\end{array}\right)& \left(\begin{array}{c}-5\\ 3\end{array}\right)& \left(\begin{array}{c}-5\\ 3\end{array}\right)\\ & & & & \left(\begin{array}{c}-5\\ 4\end{array}\right)& \left(\begin{array}{c}-5\\ 4\end{array}\right)\\ & & & & & \left(\begin{array}{c}-5\\ 5\end{array}\right)\end{array}\cdots \right\}=\left\{\begin{array}{cc}1& \begin{array}{cc}3& \begin{array}{cc}-3& -1\end{array}\end{array}\end{array}\right\}\end{array}$