= Solution
Put the generators into the rows of the integer matrix
$$
A=\begin{pmatrix}1&2&3\\2&3&1\\3&1&2\end{pmatrix}.
$$
Its determinant is $5-2-21=-18$. A full-rank integer lattice generated by its rows has index equal to the absolute determinant, hence
$$
\boxed{[L:M]=18.}
$$
For a direct verification, $u,v,e_3$ form a unimodular <basis> because their row determinant is $-1$, and $w=-7u+5v+18e_3$. In these integer coordinates $M=\mathbb Zu\oplus\mathbb Zv\oplus18\mathbb Ze_3$, visibly leaving eighteen cosets.
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