Let $M$ be a free $\mathbb{Z}$-module with basis $\{a,b,c,d\}$. Let $U=\langle w,x,y,z\rangle$ with
$w=-a+3b+2c+8d$,
$x=3b+2c+8d$,
$y=5a+b-4c+8d$,
$z=7a+4b-2c+16d$.
Find the invariant factors of $M/U$.
How do I tackle this exercise? I thought about finding the Smith Normal Form of the inclusion of U in M, but I am clueless about how to proceed from that point.
Thank you for any help in advance.
If $ U $ is the submodule generated by said elements, then we can define a $ \mathbf Z $-linear map $ f : M \to M $ whose image is $ U $, which has the matrix form
$$ \begin{bmatrix} -1 & 3 & 2 & 8 \\ 0 & 3 & 2 & 8 \\ 5 & 1 & -4 & 8 \\ 7 & 4 & -2 & 16 \end{bmatrix} $$
with respect to the given basis for $ M $. We then want to describe the cokernel of this map $ f : M \to M $, and the easiest way to do this is to choose different bases for the domain and the codomain of $ f $ so that it is in Smith normal form. Doing this gives that the cokernel is isomorphic to the quotient module $ \mathbf Z^4 / (\mathbf Z^2 \oplus 2 \mathbf Z \oplus 0) \cong \mathbf Z/2 \mathbf Z \oplus \mathbf Z $.