Tag 029A

\(N(A_{15} D_{9})\)

N(A15 D9)·NIEM_A15_D9·Niemeier lattice with root system N(A15 D9)

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

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)\(e_{13}\)\(e_{14}\)\(e_{15}\)\(e_{16}\)\(e_{17}\)\(e_{18}\)\(e_{19}\)\(e_{20}\)\(e_{21}\)\(e_{22}\)\(e_{23}\)\(e_{24}\)
\(e_{1}\)211-1-1-1-1-10000000000111111
\(e_{2}\)121-1-1-1-1-10000000000111111
\(e_{3}\)112-1-1-1-1-10000000000111111
\(e_{4}\)-1-1-12111100000000000000-1-1
\(e_{5}\)-1-1-1121110000000000-1-1-10-1-1
\(e_{6}\)-1-1-1112110000000000-1-1-1-1-1-1
\(e_{7}\)-1-1-1111210000000000-1-1-1-1-1-1
\(e_{8}\)-1-1-111112-1-1-10000000000000
\(e_{9}\)0000000-12110000000-1-1-1-1-1-1
\(e_{10}\)0000000-11210000000-1-1-1-1-1-1
\(e_{11}\)0000000-11120000000-1-1-1-1-1-1
\(e_{12}\)00000000000211-1-1-1-1-11111-1
\(e_{13}\)000000000001210-100-11110-1
\(e_{14}\)000000000001120000-11111-1
\(e_{15}\)00000000000-1002111000001
\(e_{16}\)00000000000-1-101211000001
\(e_{17}\)00000000000-10011210000-10
\(e_{18}\)00000000000-1001112000001
\(e_{19}\)1110-1-1-10-1-1-1-1-1-10000411112
\(e_{20}\)1110-1-1-10-1-1-11110000143321
\(e_{21}\)1110-1-1-10-1-1-11110000134321
\(e_{22}\)11100-1-10-1-1-11110000133421
\(e_{23}\)111-1-1-1-10-1-1-110100-10122242
\(e_{24}\)111-1-1-1-10-1-1-1-1-1-11101211124
 
[[2, 1, 1, -1, -1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1], [1, 2, 1, -1, -1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1], [-1, -1, -1, 1, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, -1, 0, -1, -1], [-1, -1, -1, 1, 1, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, -1, -1, -1, -1], [-1, -1, -1, 1, 1, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, -1, -1, -1, -1], [-1, -1, -1, 1, 1, 1, 1, 2, -1, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, -1, -1, -1, -1, -1, -1], [0, 0, 0, 0, 0, 0, 0, -1, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, -1, -1, -1, -1, -1, -1], [0, 0, 0, 0, 0, 0, 0, -1, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, -1, -1, -1, -1, -1, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 1, -1, -1, -1, -1, -1, 1, 1, 1, 1, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, -1, 0, 0, -1, 1, 1, 1, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 0, 0, 0, 0, -1, 1, 1, 1, 1, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 2, 1, 1, 1, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0, 1, 2, 1, 1, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, 1, 2, 1, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, 1, 1, 2, 0, 0, 0, 0, 0, 1], [1, 1, 1, 0, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, 0, 0, 0, 4, 1, 1, 1, 1, 2], [1, 1, 1, 0, -1, -1, -1, 0, -1, -1, -1, 1, 1, 1, 0, 0, 0, 0, 1, 4, 3, 3, 2, 1], [1, 1, 1, 0, -1, -1, -1, 0, -1, -1, -1, 1, 1, 1, 0, 0, 0, 0, 1, 3, 4, 3, 2, 1], [1, 1, 1, 0, 0, -1, -1, 0, -1, -1, -1, 1, 1, 1, 0, 0, 0, 0, 1, 3, 3, 4, 2, 1], [1, 1, 1, -1, -1, -1, -1, 0, -1, -1, -1, 1, 0, 1, 0, 0, -1, 0, 1, 2, 2, 2, 4, 2], [1, 1, 1, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 1, 1, 0, 1, 2, 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 number384

Theta series

\[ \sum_{x \in L} q^{b(x, x)} = 1 + 384q^{2} + 187344q^{4} + O(q^{5}) \]

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)no
Number of roots384
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong A_{15} \oplus D_{9}\)
Rank of \(R(L)\)24
\([L : R(L)]\)8
\(R(L)\) is primitive in \(L\)no: \(L / R(L) \cong \mathbb{Z}/8\)

Components of \(\Phi(L)\)

ComponentTypeValues of \(b(r, r)\)Number of roots\(\mathbb{Z}\Phi_c\)
\(\Phi_{1}\)\(A_{15}\)\(\{2\}\)240\(\cong A_{15}\)
\(\Phi_{2}\)\(D_{9}\)\(\{2\}\)144\(\cong D_{9}\)

Simple roots

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

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

Notes

The components \(b(e_i, e_j)\) are those of the section GRAM of the entry NIEM_A15_D9 of the Catalogue of Lattices.

References

  1. G. Nebe and N. J. A. Sloane, Catalogue of Lattices, entry NIEM_A15_D9. https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/NIEM_A15_D9.html