A typical $E_8$ lattice is of the form of $E_8$ Cartan matrix:
$$\begin{pmatrix} 2 &−1 &0 &0& 0& 0& 0& 0\\ −1& 2 &−1& 0& 0& 0& 0& 0\\ 0& −1& 2& −1& 0& 0& 0& 0\\ 0& 0& −1& 2& −1& 0& 0& 0\\ 0& 0& 0& −1& 2& −1& 0& −1\\ 0& 0& 0& 0& −1& 2& −1& 0\\ 0& 0& 0& 0& 0& −1& 2& 0\\ 0& 0& 0& 0& −1& 0& 0& 2 \end{pmatrix}$$
Which $E_8$ Cartan matrix has all integer entries and with determinant $=1$.
This is a symmetric unimodular matrix.
Question
Can you provide an explicit form of
- the rank-24 Leech lattice that is also symmetric unimodular matrix with integer entries?
What does it look like?
Note that the Leech generating matrix in Wikipedia is NOT symmetric and NOT unimodular, thus fails my criteria.
https://en.wikipedia.org/wiki/Leech_lattice#Constructions
Similar p.3 of this note shows NOT symmetric and NOT unimodular matrix: http://math.ups.edu/~bryans/Current/Journal_Spring_2006/ARoberts_LeechLattice.pdf