Tag 0294

\(E_{8}^{3}\)

E8^3·NIEM_3E8·Niemeier lattice with root system E8^3

Invariants

Rank24
Signature \((n_+, n_-)\)(24, 0)
Determinant\(1\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{24}\).

In this basis, \(L = {}\)\(\mathbb{Z}\{e_{1}, \dots, e_{8}\}\) \(\oplus\) \(\mathbb{Z}\{e_{9}, \dots, e_{16}\}\) \(\oplus\) \(\mathbb{Z}\{e_{17}, \dots, e_{24}\}\), an orthogonal sum.

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)\(e_{13}\)\(e_{14}\)\(e_{15}\)\(e_{16}\)\(e_{17}\)\(e_{18}\)\(e_{19}\)\(e_{20}\)\(e_{21}\)\(e_{22}\)\(e_{23}\)\(e_{24}\)
\(e_{1}\)211111110000000000000000
\(e_{2}\)121000110000000000000000
\(e_{3}\)112110110000000000000000
\(e_{4}\)101211110000000000000000
\(e_{5}\)101121110000000000000000
\(e_{6}\)100112110000000000000000
\(e_{7}\)111111210000000000000000
\(e_{8}\)111111120000000000000000
\(e_{9}\)0000000020-11111-100000000
\(e_{10}\)000000000201000000000000
\(e_{11}\)00000000-102-1-1-1-1100000000
\(e_{12}\)0000000011-12111-100000000
\(e_{13}\)0000000010-11211-100000000
\(e_{14}\)0000000010-11121000000000
\(e_{15}\)0000000010-11112000000000
\(e_{16}\)00000000-101-1-100200000000
\(e_{17}\)000000000000000021-1-1-1-1-1-1
\(e_{18}\)000000000000000012-10-1-1-1-1
\(e_{19}\)0000000000000000-1-1211111
\(e_{20}\)0000000000000000-10121110
\(e_{21}\)0000000000000000-1-1112111
\(e_{22}\)0000000000000000-1-1111210
\(e_{23}\)0000000000000000-1-1111120
\(e_{24}\)0000000000000000-1-1101002
 
[[2, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 2, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 1, 2, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 1, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 1, 1, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 1, 1, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 1, 1, 1, 1, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 1, 1, 1, 1, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, 0, -1, 1, 1, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 2, -1, -1, -1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, -1, 2, 1, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 1, 2, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 1, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 1, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1, -1, -1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, -1, -1, -1, -1, -1, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, -1, 0, -1, -1, -1, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 2, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1, 2, 1, 1, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 1, 1, 2, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 1, 1, 1, 2, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 1, 1, 1, 1, 2, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 1, 0, 1, 0, 0, 2]]

Integral lattice

Parityeven
Discriminant group\(0\)
Unimodularyes
Integral overlattices \(L \subseteq M \subseteq L^*\)1
Nikulin invariants \((r, a, \delta)\)\((24, 0, 0)\)
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)\(\{2\}\)
Character of the discriminant \(\chi_D\), \(D = (-1)^{12} \det L\)trivial
Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)\(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 23)\, \zeta^\Sigma(s - 12)}{\zeta^\Sigma(s - 11)}\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)\(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 23)}{\zeta^\Sigma(s - 11)}\)

Definite lattice

Minimum of \(b(x, x)\), \(x \neq 0\)\(2\)
Kissing number720

Theta series

\[ \sum_{x \in L} q^{b(x, x)} = 1 + 720q^{2} + 179280q^{4} + O(q^{5}) \]

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)yes
\(L = \mathbb{Z}\Phi_S(L)\) for\(S = \{2\}\)
Number of roots720
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong E_{8}^{3}\)
Rank of \(R(L)\)24
\([L : R(L)]\)1
\(R(L)\) is primitive in \(L\)yes

Components of \(\Phi(L)\)

ComponentTypeValues of \(b(r, r)\)Number of roots\(\mathbb{Z}\Phi_c\)
\(\Phi_{1}\)\(E_{8}\)\(\{2\}\)240\(\cong E_{8}\)
\(\Phi_{2}\)\(E_{8}\)\(\{2\}\)240\(\cong E_{8}\)
\(\Phi_{3}\)\(E_{8}\)\(\{2\}\)240\(\cong E_{8}\)

Simple roots

\(\alpha^{(c)}_j\) is simple root \(j\) of the component \(\Phi_c\).

Simple rootCoordinates\(b(r, r)\)
\(\alpha^{(1)}_{1}\)\(e_{18} - e_{19} - e_{20} + e_{22} + e_{23} + e_{24}\)\(2\)
\(\alpha^{(1)}_{2}\)\(e_{19} - e_{20} - e_{24}\)\(2\)
\(\alpha^{(1)}_{3}\)\(e_{20} - e_{21}\)\(2\)
\(\alpha^{(1)}_{4}\)\(e_{24}\)\(2\)
\(\alpha^{(1)}_{5}\)\(e_{21} - e_{22} - e_{24}\)\(2\)
\(\alpha^{(1)}_{6}\)\(e_{22} - e_{23}\)\(2\)
\(\alpha^{(1)}_{7}\)\(e_{23}\)\(2\)
\(\alpha^{(1)}_{8}\)\(e_{17} - e_{18} - e_{19} + e_{20} - e_{21} + e_{24}\)\(2\)
\(\alpha^{(2)}_{1}\)\(e_{15}\)\(2\)
\(\alpha^{(2)}_{2}\)\(e_{12} - e_{13}\)\(2\)
\(\alpha^{(2)}_{3}\)\(e_{14} - e_{15}\)\(2\)
\(\alpha^{(2)}_{4}\)\(e_{13} - e_{14}\)\(2\)
\(\alpha^{(2)}_{5}\)\(e_{16}\)\(2\)
\(\alpha^{(2)}_{6}\)\(e_{11} - e_{12} - e_{13} + e_{14} + e_{15} - 2e_{16}\)\(2\)
\(\alpha^{(2)}_{7}\)\(e_{10} - e_{11} - e_{12}\)\(2\)
\(\alpha^{(2)}_{8}\)\(e_{9} - 2e_{10} - e_{11} + 3e_{12} + e_{13} - 2e_{14} - 2e_{15} + 3e_{16}\)\(2\)
\(\alpha^{(3)}_{1}\)\(e_{2} - 2e_{3} + e_{4} + e_{5} - e_{6}\)\(2\)
\(\alpha^{(3)}_{2}\)\(e_{4} - e_{5}\)\(2\)
\(\alpha^{(3)}_{3}\)\(e_{3} - e_{4} - e_{5} + e_{6}\)\(2\)
\(\alpha^{(3)}_{4}\)\(e_{5} - e_{6}\)\(2\)
\(\alpha^{(3)}_{5}\)\(e_{6} - e_{7}\)\(2\)
\(\alpha^{(3)}_{6}\)\(e_{7} - e_{8}\)\(2\)
\(\alpha^{(3)}_{7}\)\(e_{8}\)\(2\)
\(\alpha^{(3)}_{8}\)\(e_{1} - 2e_{2} - e_{4} - e_{5} - e_{6} + e_{7} + e_{8}\)\(2\)

Notes

The components \(b(e_i, e_j)\) are those of the section GRAM of the entry NIEM_3E8 of the Catalogue of Lattices.

References

  1. G. Nebe and N. J. A. Sloane, Catalogue of Lattices, entry NIEM_3E8. https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/NIEM_3E8.html