Tag 028X

\(N(D_{4}^{6})\)

N(D4^6)·NIEM_6D4·Niemeier lattice with root system N(D4^6)

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

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)\(e_{13}\)\(e_{14}\)\(e_{15}\)\(e_{16}\)\(e_{17}\)\(e_{18}\)\(e_{19}\)\(e_{20}\)\(e_{21}\)\(e_{22}\)\(e_{23}\)\(e_{24}\)
\(e_{1}\)2010000000000-10-111111110
\(e_{2}\)0210000000000-1-1-101100-1-10
\(e_{3}\)1120000000000-1-1-111111000
\(e_{4}\)0002110000000-11-11-1-11-1101
\(e_{5}\)0001210000000-1101-1-11-1001
\(e_{6}\)0001120000000-1001-1-100100
\(e_{7}\)000000200000000000000111
\(e_{8}\)0000000201100-1011000010-1
\(e_{9}\)000000002-1-100010-1-1-11-1-110
\(e_{10}\)00000001-12100-1-1011100100
\(e_{11}\)00000001-11200-1-10111-111-1-1
\(e_{12}\)000000000002100001-11-1011
\(e_{13}\)000000000001200001-11-1000
\(e_{14}\)-1-1-1-1-1-10-10-1-100401-1-1-1-10-100
\(e_{15}\)0-1-1110001-1-100041-1-2-21-2011
\(e_{16}\)-1-1-1-1000100000114-1-1-1-1000-1
\(e_{17}\)10111101-11100-1-1-140011200
\(e_{18}\)111-1-1-100-11111-1-2-104201000
\(e_{19}\)111-1-1-100-111-1-1-1-2-1024-120-1-1
\(e_{20}\)1011100010-111-11-110-14-2022
\(e_{21}\)101-1-1000-101-1-10-20112-241-1-2
\(e_{22}\)1-1010111-11100-10020001410
\(e_{23}\)1-100001010-11001000-12-1142
\(e_{24}\)0001101-100-11001-100-12-2024
 
[[2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, 1, 1, 1, 1, 1, 1, 1, 0], [0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, -1, 0, 1, 1, 0, 0, -1, -1, 0], [1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, -1, 1, 1, 1, 1, 1, 0, 0, 0], [0, 0, 0, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, -1, 1, -1, 1, -1, -1, 1, -1, 1, 0, 1], [0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 1, -1, -1, 1, -1, 0, 0, 1], [0, 0, 0, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, -1, -1, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 1, 0, 0, -1, 0, 1, 1, 0, 0, 0, 0, 1, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 2, -1, -1, 0, 0, 0, 1, 0, -1, -1, -1, 1, -1, -1, 1, 0], [0, 0, 0, 0, 0, 0, 0, 1, -1, 2, 1, 0, 0, -1, -1, 0, 1, 1, 1, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, -1, 1, 2, 0, 0, -1, -1, 0, 1, 1, 1, -1, 1, 1, -1, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 1, -1, 1, -1, 0, 0, 0], [-1, -1, -1, -1, -1, -1, 0, -1, 0, -1, -1, 0, 0, 4, 0, 1, -1, -1, -1, -1, 0, -1, 0, 0], [0, -1, -1, 1, 1, 0, 0, 0, 1, -1, -1, 0, 0, 0, 4, 1, -1, -2, -2, 1, -2, 0, 1, 1], [-1, -1, -1, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 4, -1, -1, -1, -1, 0, 0, 0, -1], [1, 0, 1, 1, 1, 1, 0, 1, -1, 1, 1, 0, 0, -1, -1, -1, 4, 0, 0, 1, 1, 2, 0, 0], [1, 1, 1, -1, -1, -1, 0, 0, -1, 1, 1, 1, 1, -1, -2, -1, 0, 4, 2, 0, 1, 0, 0, 0], [1, 1, 1, -1, -1, -1, 0, 0, -1, 1, 1, -1, -1, -1, -2, -1, 0, 2, 4, -1, 2, 0, -1, -1], [1, 0, 1, 1, 1, 0, 0, 0, 1, 0, -1, 1, 1, -1, 1, -1, 1, 0, -1, 4, -2, 0, 2, 2], [1, 0, 1, -1, -1, 0, 0, 0, -1, 0, 1, -1, -1, 0, -2, 0, 1, 1, 2, -2, 4, 1, -1, -2], [1, -1, 0, 1, 0, 1, 1, 1, -1, 1, 1, 0, 0, -1, 0, 0, 2, 0, 0, 0, 1, 4, 1, 0], [1, -1, 0, 0, 0, 0, 1, 0, 1, 0, -1, 1, 0, 0, 1, 0, 0, 0, -1, 2, -1, 1, 4, 2], [0, 0, 0, 1, 1, 0, 1, -1, 0, 0, -1, 1, 0, 0, 1, -1, 0, 0, -1, 2, -2, 0, 2, 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 number144

Theta series

\[ \sum_{x \in L} q^{b(x, x)} = 1 + 144q^{2} + 193104q^{4} + O(q^{5}) \]

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)no
Number of roots144
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong D_{4}^{6}\)
Rank of \(R(L)\)24
\([L : R(L)]\)64
\(R(L)\) is primitive in \(L\)no: \(L / R(L) \cong (\mathbb{Z}/2)^{6}\)

Components of \(\Phi(L)\)

ComponentTypeValues of \(b(r, r)\)Number of roots\(\mathbb{Z}\Phi_c\)
\(\Phi_{1}\)\(D_{4}\)\(\{2\}\)24\(\cong D_{4}\)
\(\Phi_{2}\)\(D_{4}\)\(\{2\}\)24\(\cong D_{4}\)
\(\Phi_{3}\)\(D_{4}\)\(\{2\}\)24\(\cong D_{4}\)
\(\Phi_{4}\)\(D_{4}\)\(\{2\}\)24\(\cong D_{4}\)
\(\Phi_{5}\)\(D_{4}\)\(\{2\}\)24\(\cong D_{4}\)
\(\Phi_{6}\)\(D_{4}\)\(\{2\}\)24\(\cong D_{4}\)

Simple roots

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

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

Notes

The components \(b(e_i, e_j)\) are those of the section GRAM of the entry NIEM_6D4 of the Catalogue of Lattices.

References

  1. G. Nebe and N. J. A. Sloane, Catalogue of Lattices, entry NIEM_6D4. https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/NIEM_6D4.html