Tag 0290

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

N(D6^4)·NIEM_4D6·Niemeier lattice with root system N(D6^4)

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

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)\(e_{13}\)\(e_{14}\)\(e_{15}\)\(e_{16}\)\(e_{17}\)\(e_{18}\)\(e_{19}\)\(e_{20}\)\(e_{21}\)\(e_{22}\)\(e_{23}\)\(e_{24}\)
\(e_{1}\)200000000000000111111111
\(e_{2}\)02000000000000-11111111-10
\(e_{3}\)002000000000001-1-1-1-1-1-1-10-1
\(e_{4}\)00020000000000011-1-100000
\(e_{5}\)00002-1-1-1-100000-100011-1-110
\(e_{6}\)0000-12111000001000-1-111-10
\(e_{7}\)0000-11211000000000-1-11101
\(e_{8}\)0000-11121-100001000-1-111-11
\(e_{9}\)0000-11112-100000000001101
\(e_{10}\)0000000-1-120000000000000-1
\(e_{11}\)000000000020111-1-110010-1-1
\(e_{12}\)000000000002111-1-110001-1-1
\(e_{13}\)000000000011211-1-110011-1-1
\(e_{14}\)000000000011121-1-110011-1-1
\(e_{15}\)0-110-11010011114-2-20-2-200-2-2
\(e_{16}\)11-11000000-1-1-1-1-2430121112
\(e_{17}\)11-11000000-1-1-1-1-2340121112
\(e_{18}\)11-1-1000000111100042222-10
\(e_{19}\)11-1-11-1-1-1000000-2112431111
\(e_{20}\)11-101-1-1-1000000-2222341111
\(e_{21}\)11-10-111110101101121143-11
\(e_{22}\)11-10-111110011101121134-11
\(e_{23}\)1-1001-10-100-1-1-1-1-211-111-1-142
\(e_{24}\)10-1000111-1-1-1-1-1-2220111124
 
[[2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1], [0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 1, 1, 1, 1, 1, 1, -1, 0], [0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, -1, -1, -1, -1, -1, -1, 0, -1], [0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, -1, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 2, -1, -1, -1, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 1, 1, -1, -1, 1, 0], [0, 0, 0, 0, -1, 2, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1, -1, 1, 1, -1, 0], [0, 0, 0, 0, -1, 1, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 1, 1, 0, 1], [0, 0, 0, 0, -1, 1, 1, 2, 1, -1, 0, 0, 0, 0, 1, 0, 0, 0, -1, -1, 1, 1, -1, 1], [0, 0, 0, 0, -1, 1, 1, 1, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1], [0, 0, 0, 0, 0, 0, 0, -1, -1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 1, 1, -1, -1, 1, 0, 0, 1, 0, -1, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 1, 1, -1, -1, 1, 0, 0, 0, 1, -1, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 1, 1, -1, -1, 1, 0, 0, 1, 1, -1, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 1, -1, -1, 1, 0, 0, 1, 1, -1, -1], [0, -1, 1, 0, -1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 4, -2, -2, 0, -2, -2, 0, 0, -2, -2], [1, 1, -1, 1, 0, 0, 0, 0, 0, 0, -1, -1, -1, -1, -2, 4, 3, 0, 1, 2, 1, 1, 1, 2], [1, 1, -1, 1, 0, 0, 0, 0, 0, 0, -1, -1, -1, -1, -2, 3, 4, 0, 1, 2, 1, 1, 1, 2], [1, 1, -1, -1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 4, 2, 2, 2, 2, -1, 0], [1, 1, -1, -1, 1, -1, -1, -1, 0, 0, 0, 0, 0, 0, -2, 1, 1, 2, 4, 3, 1, 1, 1, 1], [1, 1, -1, 0, 1, -1, -1, -1, 0, 0, 0, 0, 0, 0, -2, 2, 2, 2, 3, 4, 1, 1, 1, 1], [1, 1, -1, 0, -1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 2, 1, 1, 4, 3, -1, 1], [1, 1, -1, 0, -1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 2, 1, 1, 3, 4, -1, 1], [1, -1, 0, 0, 1, -1, 0, -1, 0, 0, -1, -1, -1, -1, -2, 1, 1, -1, 1, 1, -1, -1, 4, 2], [1, 0, -1, 0, 0, 0, 1, 1, 1, -1, -1, -1, -1, -1, -2, 2, 2, 0, 1, 1, 1, 1, 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 number240

Theta series

\[ \sum_{x \in L} q^{b(x, x)} = 1 + 240q^{2} + 190800q^{4} + O(q^{5}) \]

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)no
Number of roots240
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong D_{6}^{4}\)
Rank of \(R(L)\)24
\([L : R(L)]\)16
\(R(L)\) is primitive in \(L\)no: \(L / R(L) \cong (\mathbb{Z}/2)^{4}\)

Components of \(\Phi(L)\)

ComponentTypeValues of \(b(r, r)\)Number of roots\(\mathbb{Z}\Phi_c\)
\(\Phi_{1}\)\(D_{6}\)\(\{2\}\)60\(\cong D_{6}\)
\(\Phi_{2}\)\(D_{6}\)\(\{2\}\)60\(\cong D_{6}\)
\(\Phi_{3}\)\(D_{6}\)\(\{2\}\)60\(\cong D_{6}\)
\(\Phi_{4}\)\(D_{6}\)\(\{2\}\)60\(\cong D_{6}\)

Simple roots

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

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

Notes

The components \(b(e_i, e_j)\) are those of the section GRAM of the entry NIEM_4D6 of the Catalogue of Lattices.

References

  1. G. Nebe and N. J. A. Sloane, Catalogue of Lattices, entry NIEM_4D6. https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/NIEM_4D6.html