For $n$ an integer divisible by $8$, let me denote by $E\_n$ the "usual" even non-degenerate positive definite integral symmetric bilinear form over $\mathbf Z^n$.


It is well known that in dimension 16, the only even non-degenerate positive definite integral symmetric bilinear form are (up to isometry) $E\_8 \oplus E\_8$ and $E\_{16}$. Moreover, it is known that these two have same theta series (i.e. their quadratic forms represent the same integers, the same amount of times).


My question is thus: given a dimension 16 non-degenerate positive definite integral symmetric bilinear form, what is the easiest test to determine whether it is isometric to $E\_8 \oplus E\_8$ or to $E\_{16}$?



Easiest way to distinguish $E_8 \oplus E_8$ from $E_{16}$
One of the possible tricks I was after was the following: the lattice generated by the roots (vectors of length $2$) of $E\_8 \oplus E\_8$ is $E\_8 \oplus E\_8$, whereas the lattice generated by the roots of $E\_{16}$ is $D\_{16}$ (which is of index $2$ in $E\_{16}$).


Pari/gp has a very efficient way to list the roots. Putting the resulting matrix in Hermite normal form lets me see whether the index is $1$ or $2$.


