+ b\sqrt{2} - c\sqrt{3} - d\sqrt{6}.
  \end{align*}

  Therefore

  \[
    \tau \phi_3 \tau^{-1} = \phi_3.
  \]

  We have shown that

  \begin{equation}
    \tau M^* \tau^{-1} = \{ \phi_1, \phi_3 \}.
    \label{eq-tau-coset}
  \end{equation}

  From \( \eqnref{eq-tau-m-ast}{1} \) and \( \eqnref{eq-tau-coset}{2} \)
  we see that

  \[
     \tau M^* \tau^{-1} = \tau(M)^*.
  \]

  Since we are working with a concrete example of \( \tau \) here, we
  know exactly how it behaves, so we succeeded in demonstratin