I am making a latex beamer presentation and I have a table. My table contains equation in math modes and looks like

enter image description here

In the last part, Factorization, I want to align the second Delta of chi twelve with the one from psi twelve from the last line. When this is done, I want to align the arrow from chi ten with the arrow from chi10. I tried many thing but nothing worked out right now. Here is my code


\frametitle{Siegel modular forms}

 \begin{tabu}{|c| c| c|}
 \hline %inserts horizontal line
 &Genus 1 &  Genus 2 \\[0.5ex] 
 Generators  & $\phi_4$, $\phi_6$ & $\psi_4,\psi_6,\chi_{10},\chi_{12}$ \\
 Siegel operator & $\Phi(\phi_4) = \Phi(\phi_6) = 1 $& $\Phi(\psi_i)=\phi_i,      \Phi(\chi_{j})= 0$\\
 Discriminant & $\Delta =\frac{\phi_4^3-\phi_6^2}{1728} = \eta^{24}$ &   $\psi_{12} = \frac{\psi_4^3-\psi_6^2}{1728}$ 
  Siegel operator & $\Phi(\Delta)=0$ $\rightarrow$ cusp form &   $\Phi(\psi_{12})=\Delta$\\ \hline  
Factorization & \multicolumn{2}{c|}{$\begin{array} {lcllcl} \psi_4 &    \rightarrow &\phi_4\otimes\phi_4 &\chi_{10}&\rightarrow &0\\ \psi_6 &   \rightarrow & \phi_6\otimes\phi_6 & \chi_{12} & \rightarrow &   \Delta\otimes\Delta\\
 \psi_{12} & \rightarrow &  \multicolumn{4}{l}{\phi_4^3\otimes \Delta +  \Delta \otimes \phi_4^3-1728\Delta\otimes\Delta }


Thank you very much.


enter image description here

    \psi_4    & \rightarrow \phi_4\otimes\phi_4 & \chi_{10}  \rightarrow {} & 0\\
    \psi_6    & \rightarrow \phi_6\otimes\phi_6 & \chi_{12}  \rightarrow {} & \Delta\otimes\Delta\\
    \psi_{12} & \multicolumn{2}{@{}l@{}}{\rightarrow \phi_4^3\otimes \Delta +
                                              \Delta \otimes \phi_4^3-1728} & \Delta\otimes\Delta

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