Tag 0296

\(N(A_{9}^{2} D_{6})\)

N(A9^2 D6)·NIEM_2A9_D6·Niemeier lattice with root system N(A9^2 D6)

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

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)\(e_{13}\)\(e_{14}\)\(e_{15}\)\(e_{16}\)\(e_{17}\)\(e_{18}\)\(e_{19}\)\(e_{20}\)\(e_{21}\)\(e_{22}\)\(e_{23}\)\(e_{24}\)
\(e_{1}\)2-10000000000000110100000
\(e_{2}\)-121-1-10000000000001011111
\(e_{3}\)012-1-10000000000101111111
\(e_{4}\)0-1-1210000000000-1-1-1-1-1-1-1-1-1
\(e_{5}\)0-1-1120000000000-1-1-1-1-1-1-1-1-1
\(e_{6}\)0000020000000001-1-1000000
\(e_{7}\)000000211-1-1-10000-1000-1000
\(e_{8}\)000000121-1-1-10000-1-1-10-1000
\(e_{9}\)000000112-1-1-10001-1-1-1-10000
\(e_{10}\)000000-1-1-1211-100010001-1-1-1
\(e_{11}\)000000-1-1-1121000000001-1-1-1
\(e_{12}\)000000-1-1-1112010-111111-1-1-1
\(e_{13}\)000000000-10020000110-1111
\(e_{14}\)000000000001020-101110-1-1-1
\(e_{15}\)00000000000000200001000-1
\(e_{16}\)101-1-1100100-10-1040-10-11111
\(e_{17}\)100-1-1-1-1-1-1101000042211111
\(e_{18}\)011-1-1-10-1-1001110-124320111
\(e_{19}\)101-1-100-1-1001110023420111
\(e_{20}\)011-1-1000-1001011-112240000
\(e_{21}\)011-1-10-1-10111-100110004000
\(e_{22}\)011-1-10000-1-1-11-10111100433
\(e_{23}\)011-1-10000-1-1-11-10111100343
\(e_{24}\)011-1-10000-1-1-11-1-1111100334
 
[[2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0], [-1, 2, 1, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1], [0, 1, 2, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1], [0, -1, -1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, -1, -1, -1, -1, -1, -1, -1], [0, -1, -1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, -1, -1, -1, -1, -1, -1, -1], [0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2, 1, 1, -1, -1, -1, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 2, 1, -1, -1, -1, 0, 0, 0, 0, -1, -1, -1, 0, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 1, 2, -1, -1, -1, 0, 0, 0, 1, -1, -1, -1, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, -1, -1, 2, 1, 1, -1, 0, 0, 0, 1, 0, 0, 0, 1, -1, -1, -1], [0, 0, 0, 0, 0, 0, -1, -1, -1, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, -1, -1], [0, 0, 0, 0, 0, 0, -1, -1, -1, 1, 1, 2, 0, 1, 0, -1, 1, 1, 1, 1, 1, -1, -1, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 2, 0, 0, 0, 0, 1, 1, 0, -1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, -1, 0, 1, 1, 1, 0, -1, -1, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, -1], [1, 0, 1, -1, -1, 1, 0, 0, 1, 0, 0, -1, 0, -1, 0, 4, 0, -1, 0, -1, 1, 1, 1, 1], [1, 0, 0, -1, -1, -1, -1, -1, -1, 1, 0, 1, 0, 0, 0, 0, 4, 2, 2, 1, 1, 1, 1, 1], [0, 1, 1, -1, -1, -1, 0, -1, -1, 0, 0, 1, 1, 1, 0, -1, 2, 4, 3, 2, 0, 1, 1, 1], [1, 0, 1, -1, -1, 0, 0, -1, -1, 0, 0, 1, 1, 1, 0, 0, 2, 3, 4, 2, 0, 1, 1, 1], [0, 1, 1, -1, -1, 0, 0, 0, -1, 0, 0, 1, 0, 1, 1, -1, 1, 2, 2, 4, 0, 0, 0, 0], [0, 1, 1, -1, -1, 0, -1, -1, 0, 1, 1, 1, -1, 0, 0, 1, 1, 0, 0, 0, 4, 0, 0, 0], [0, 1, 1, -1, -1, 0, 0, 0, 0, -1, -1, -1, 1, -1, 0, 1, 1, 1, 1, 0, 0, 4, 3, 3], [0, 1, 1, -1, -1, 0, 0, 0, 0, -1, -1, -1, 1, -1, 0, 1, 1, 1, 1, 0, 0, 3, 4, 3], [0, 1, 1, -1, -1, 0, 0, 0, 0, -1, -1, -1, 1, -1, -1, 1, 1, 1, 1, 0, 0, 3, 3, 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
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong A_{9}^{2} \oplus D_{6}\)
Rank of \(R(L)\)24
\([L : R(L)]\)20
\(R(L)\) is primitive in \(L\)no: \(L / R(L) \cong \mathbb{Z}/2 \oplus \mathbb{Z}/10\)

Components of \(\Phi(L)\)

ComponentTypeValues of \(b(r, r)\)Number of roots\(\mathbb{Z}\Phi_c\)
\(\Phi_{1}\)\(A_{9}\)\(\{2\}\)90\(\cong A_{9}\)
\(\Phi_{2}\)\(A_{9}\)\(\{2\}\)90\(\cong A_{9}\)
\(\Phi_{3}\)\(D_{6}\)\(\{2\}\)60\(\cong D_{6}\)

Simple roots

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

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

Notes

The components \(b(e_i, e_j)\) are those of the section GRAM of the entry NIEM_2A9_D6 of the Catalogue of Lattices.

References

  1. G. Nebe and N. J. A. Sloane, Catalogue of Lattices, entry NIEM_2A9_D6. https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/NIEM_2A9_D6.html