Tag 0293

\(N(D_{8}^{3})\)

N(D8^3)·NIEM_3D8·Niemeier lattice with root system N(D8^3)

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 \(e_{1}, \dots, e_{24}\).

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)\(e_{13}\)\(e_{14}\)\(e_{15}\)\(e_{16}\)\(e_{17}\)\(e_{18}\)\(e_{19}\)\(e_{20}\)\(e_{21}\)\(e_{22}\)\(e_{23}\)\(e_{24}\)
\(e_{1}\)211111000000000001110011
\(e_{2}\)121111-10000000000111-1-111
\(e_{3}\)112111000000000001100-111
\(e_{4}\)11121100000000000001-1011
\(e_{5}\)111121-10000000000001-1011
\(e_{6}\)111112000000000001100-111
\(e_{7}\)0-100-102000000000000-11000
\(e_{8}\)00000002-1101100001-1-1011-1
\(e_{9}\)0000000-12-10-1-10000-1110-1-11
\(e_{10}\)00000001-12111000010-1011-1
\(e_{11}\)000000000120000001100000
\(e_{12}\)00000001-11020000010-1011-1
\(e_{13}\)00000001-1100200000-1-1011-1
\(e_{14}\)00000000000002111001-1-1-11
\(e_{15}\)000000000000012110010-100
\(e_{16}\)00000000000001121001-1-1-11
\(e_{17}\)000000000000011120010-1-10
\(e_{18}\)11100101-11110000042-11-110
\(e_{19}\)1110010-11010-100002401-201
\(e_{20}\)110110-1-11-10-1-11111-104-2-1-12
\(e_{21}\)0-10-1-101000000-10-1011-2400-2
\(e_{22}\)0-1-100-101-11011-1-1-1-1-1-2-1041-2
\(e_{23}\)11111101-11011-10-1-110-1014-1
\(e_{24}\)1111110-11-10-1-11010012-2-2-14
 
[[2, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1], [1, 2, 1, 1, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, -1, -1, 1, 1], [1, 1, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, -1, 1, 1], [1, 1, 1, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, 1], [1, 1, 1, 1, 2, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, 1], [1, 1, 1, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, -1, 1, 1], [0, -1, 0, 0, -1, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 2, -1, 1, 0, 1, 1, 0, 0, 0, 0, 1, -1, -1, 0, 1, 1, -1], [0, 0, 0, 0, 0, 0, 0, -1, 2, -1, 0, -1, -1, 0, 0, 0, 0, -1, 1, 1, 0, -1, -1, 1], [0, 0, 0, 0, 0, 0, 0, 1, -1, 2, 1, 1, 1, 0, 0, 0, 0, 1, 0, -1, 0, 1, 1, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, -1, 1, 0, 2, 0, 0, 0, 0, 0, 1, 0, -1, 0, 1, 1, -1], [0, 0, 0, 0, 0, 0, 0, 1, -1, 1, 0, 0, 2, 0, 0, 0, 0, 0, -1, -1, 0, 1, 1, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 1, 1, 0, 0, 1, -1, -1, -1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 1, 0, 0, 1, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 1, 0, 0, 1, -1, -1, -1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 0, 1, 0, -1, -1, 0], [1, 1, 1, 0, 0, 1, 0, 1, -1, 1, 1, 1, 0, 0, 0, 0, 0, 4, 2, -1, 1, -1, 1, 0], [1, 1, 1, 0, 0, 1, 0, -1, 1, 0, 1, 0, -1, 0, 0, 0, 0, 2, 4, 0, 1, -2, 0, 1], [1, 1, 0, 1, 1, 0, -1, -1, 1, -1, 0, -1, -1, 1, 1, 1, 1, -1, 0, 4, -2, -1, -1, 2], [0, -1, 0, -1, -1, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, -1, 0, 1, 1, -2, 4, 0, 0, -2], [0, -1, -1, 0, 0, -1, 0, 1, -1, 1, 0, 1, 1, -1, -1, -1, -1, -1, -2, -1, 0, 4, 1, -2], [1, 1, 1, 1, 1, 1, 0, 1, -1, 1, 0, 1, 1, -1, 0, -1, -1, 1, 0, -1, 0, 1, 4, -1], [1, 1, 1, 1, 1, 1, 0, -1, 1, -1, 0, -1, -1, 1, 0, 1, 0, 0, 1, 2, -2, -2, -1, 4]]

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 number336

Theta series

\[ \sum_{x \in L} q^{b(x, x)} = 1 + 336q^{2} + 188496q^{4} + O(q^{5}) \]

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)no
Number of roots336
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong D_{8}^{3}\)
Rank of \(R(L)\)24
\([L : R(L)]\)8
\(R(L)\) is primitive in \(L\)no: \(L / R(L) \cong (\mathbb{Z}/2)^{3}\)

Components of \(\Phi(L)\)

ComponentTypeValues of \(b(r, r)\)Number of roots\(\mathbb{Z}\Phi_c\)
\(\Phi_{1}\)\(D_{8}\)\(\{2\}\)112\(\cong D_{8}\)
\(\Phi_{2}\)\(D_{8}\)\(\{2\}\)112\(\cong D_{8}\)
\(\Phi_{3}\)\(D_{8}\)\(\{2\}\)112\(\cong D_{8}\)

Simple roots

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

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

Notes

The components \(b(e_i, e_j)\) are those of the section GRAM of the entry NIEM_3D8 of the Catalogue of Lattices.

References

  1. G. Nebe and N. J. A. Sloane, Catalogue of Lattices, entry NIEM_3D8. https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/NIEM_3D8.html