Tag 029F

\(D_{24}^{+}\)

D24+·NIEM_D24·Niemeier lattice with root system D24+

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

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)\(e_{13}\)\(e_{14}\)\(e_{15}\)\(e_{16}\)\(e_{17}\)\(e_{18}\)\(e_{19}\)\(e_{20}\)\(e_{21}\)\(e_{22}\)\(e_{23}\)\(e_{24}\)
\(e_{1}\)211111111000000000000000
\(e_{2}\)121111111000000000000000
\(e_{3}\)112111111000000000000000
\(e_{4}\)111211111000000000000000
\(e_{5}\)111120111-11111-1111111100
\(e_{6}\)1111021111-1-1-1-11-1-1-1-1-1-1-111
\(e_{7}\)111111211000000000000011
\(e_{8}\)111111121000000000000011
\(e_{9}\)111111112000000000000011
\(e_{10}\)0000-110002-1-1-1-11-1-1-1-1-1-1-111
\(e_{11}\)00001-1000-12111-1111111100
\(e_{12}\)00001-1000-11211-1111111100
\(e_{13}\)00001-1000-11121-1111111100
\(e_{14}\)00001-1000-11112-1111111100
\(e_{15}\)0000-110001-1-1-1-12-1-1-1-1-1-1-111
\(e_{16}\)00001-1000-11111-12111111-10
\(e_{17}\)00001-1000-11111-11211111-1-1
\(e_{18}\)00001-1000-11111-11121111-1-1
\(e_{19}\)00001-1000-11111-1111211100
\(e_{20}\)00001-1000-11111-11111211-1-1
\(e_{21}\)00001-1000-11111-1111112100
\(e_{22}\)00001-1000-11111-11111112-1-1
\(e_{23}\)000001111100001-1-1-10-10-165
\(e_{24}\)0000011111000010-1-10-10-156
 
[[2, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 2, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 1, 2, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 1, 1, 2, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 1, 1, 1, 2, 0, 1, 1, 1, -1, 1, 1, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 0, 0], [1, 1, 1, 1, 0, 2, 1, 1, 1, 1, -1, -1, -1, -1, 1, -1, -1, -1, -1, -1, -1, -1, 1, 1], [1, 1, 1, 1, 1, 1, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, -1, 1, 0, 0, 0, 2, -1, -1, -1, -1, 1, -1, -1, -1, -1, -1, -1, -1, 1, 1], [0, 0, 0, 0, 1, -1, 0, 0, 0, -1, 2, 1, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 0, 0], [0, 0, 0, 0, 1, -1, 0, 0, 0, -1, 1, 2, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 0, 0], [0, 0, 0, 0, 1, -1, 0, 0, 0, -1, 1, 1, 2, 1, -1, 1, 1, 1, 1, 1, 1, 1, 0, 0], [0, 0, 0, 0, 1, -1, 0, 0, 0, -1, 1, 1, 1, 2, -1, 1, 1, 1, 1, 1, 1, 1, 0, 0], [0, 0, 0, 0, -1, 1, 0, 0, 0, 1, -1, -1, -1, -1, 2, -1, -1, -1, -1, -1, -1, -1, 1, 1], [0, 0, 0, 0, 1, -1, 0, 0, 0, -1, 1, 1, 1, 1, -1, 2, 1, 1, 1, 1, 1, 1, -1, 0], [0, 0, 0, 0, 1, -1, 0, 0, 0, -1, 1, 1, 1, 1, -1, 1, 2, 1, 1, 1, 1, 1, -1, -1], [0, 0, 0, 0, 1, -1, 0, 0, 0, -1, 1, 1, 1, 1, -1, 1, 1, 2, 1, 1, 1, 1, -1, -1], [0, 0, 0, 0, 1, -1, 0, 0, 0, -1, 1, 1, 1, 1, -1, 1, 1, 1, 2, 1, 1, 1, 0, 0], [0, 0, 0, 0, 1, -1, 0, 0, 0, -1, 1, 1, 1, 1, -1, 1, 1, 1, 1, 2, 1, 1, -1, -1], [0, 0, 0, 0, 1, -1, 0, 0, 0, -1, 1, 1, 1, 1, -1, 1, 1, 1, 1, 1, 2, 1, 0, 0], [0, 0, 0, 0, 1, -1, 0, 0, 0, -1, 1, 1, 1, 1, -1, 1, 1, 1, 1, 1, 1, 2, -1, -1], [0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, -1, -1, -1, 0, -1, 0, -1, 6, 5], [0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, -1, -1, 0, -1, 0, -1, 5, 6]]

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 number1104

Theta series

\[ \sum_{x \in L} q^{b(x, x)} = 1 + 1104q^{2} + 170064q^{4} + O(q^{5}) \]

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)no
Number of roots1104
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong D_{24}\)
Rank of \(R(L)\)24
\([L : R(L)]\)2
\(R(L)\) is primitive in \(L\)no: \(L / R(L) \cong \mathbb{Z}/2\)

Components of \(\Phi(L)\)

ComponentTypeValues of \(b(r, r)\)Number of roots\(\mathbb{Z}\Phi_c\)
\(\Phi_{1}\)\(D_{24}\)\(\{2\}\)1104\(\cong D_{24}\)

Simple roots

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

Notes

The components \(b(e_i, e_j)\) are those of the section GRAM of the entry NIEM_D24 of the Catalogue of Lattices.

References

  1. G. Nebe and N. J. A. Sloane, Catalogue of Lattices, entry NIEM_D24. https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/NIEM_D24.html