TITLE: What's in the genus of the cubic lattice?
QUESTION [8 upvotes]: I'll write $\mathbf{Z}^n$ for the integral quadratic form $x_1^2 + \cdots + x_n^2$.  For which values of $n$ is $\mathbf{Z}^n$ unique in its genus, i.e. isolated in Kneser's graph?  In particular can someone verify or shoot down the following guesses:

For $n \leq 8$, any lattice in the same genus as $\mathbf{Z}^n$ is isomorphic to $\mathbf{Z}^n$.
Any lattice in the same genus as $\mathbf{Z}^9$ is isomorphic to either $\mathbf{Z}^9$ or to $\mathbf{Z} \times \mathrm{E}_8$.

REPLY [3 votes]: Both statements are well-known in the arithmetic theory of quadratic forms (aka integral lattices).  
The first statement for $n \leq 5$ is a consequence of Hermite's bound on the nonzero minimum of an integral lattice: the minimum of a unimodular lattice of rank $\leq 5$ is 1.  
Both statements are  consequences of Kneser's theory of neighbors.  You can find it in the last theorem in O'Meara's book.  It also mentioned that Kneser completed the analysis of the genus of $I_n$ for $n \leq 13$.