Tag 0292

\(N(A_{8}^{3})\)

N(A8^3)·NIEM_3A8·Niemeier lattice with root system N(A8^3)

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

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)\(e_{13}\)\(e_{14}\)\(e_{15}\)\(e_{16}\)\(e_{17}\)\(e_{18}\)\(e_{19}\)\(e_{20}\)\(e_{21}\)\(e_{22}\)\(e_{23}\)\(e_{24}\)
\(e_{1}\)2-10000000000000111110011
\(e_{2}\)-12000-1000000000-1-1-1-1-111-1-1
\(e_{3}\)0021000000000001-1-1-1-10000
\(e_{4}\)0012-100000000000-1-1-1-10111
\(e_{5}\)000-121-100000000111111-1-1-1
\(e_{6}\)0-10012-100000000111110-100
\(e_{7}\)0000-1-1200000000-10-1-1-1-1000
\(e_{8}\)00000002-100000000-100110-1
\(e_{9}\)0000000-12-1-1000011100-1-111
\(e_{10}\)00000000-1210000-1-1-10010-10
\(e_{11}\)00000000-1120000-1-100000-10
\(e_{12}\)00000000000211-1-110111111
\(e_{13}\)000000000001210-100001110
\(e_{14}\)000000000001120-100111110
\(e_{15}\)00000000000-10020-1000000-1
\(e_{16}\)1-11011-101-1-1-1-1-1041100-1-200
\(e_{17}\)1-1-1-111001-1-1100-1142220-111
\(e_{18}\)1-1-1-111-1-11-10000012422-1-111
\(e_{19}\)1-1-1-111-100001010022431011
\(e_{20}\)1-1-1-111-100001010022341011
\(e_{21}\)010010-11-1101110-10-111420-1
\(e_{22}\)0101-1-101-1001110-2-1-1002410
\(e_{23}\)1-101-10001-1-11110011110142
\(e_{24}\)1-101-100-1100100-101111-1024
 
[[2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1], [-1, 2, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, -1, -1, -1, 1, 1, -1, -1], [0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, -1, -1, -1, 0, 0, 0, 0], [0, 0, 1, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, -1, -1, 0, 1, 1, 1], [0, 0, 0, -1, 2, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, -1, -1, -1], [0, -1, 0, 0, 1, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, -1, 0, 0], [0, 0, 0, 0, -1, -1, 2, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, -1, -1, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, 1, 0, -1], [0, 0, 0, 0, 0, 0, 0, -1, 2, -1, -1, 0, 0, 0, 0, 1, 1, 1, 0, 0, -1, -1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, -1, 2, 1, 0, 0, 0, 0, -1, -1, -1, 0, 0, 1, 0, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 2, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 1, -1, -1, 1, 0, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, -1, 0, 0, 0, 0, 1, 1, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 0, -1, 0, 0, 1, 1, 1, 1, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, 0, 0, 0, 0, -1], [1, -1, 1, 0, 1, 1, -1, 0, 1, -1, -1, -1, -1, -1, 0, 4, 1, 1, 0, 0, -1, -2, 0, 0], [1, -1, -1, -1, 1, 1, 0, 0, 1, -1, -1, 1, 0, 0, -1, 1, 4, 2, 2, 2, 0, -1, 1, 1], [1, -1, -1, -1, 1, 1, -1, -1, 1, -1, 0, 0, 0, 0, 0, 1, 2, 4, 2, 2, -1, -1, 1, 1], [1, -1, -1, -1, 1, 1, -1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 2, 4, 3, 1, 0, 1, 1], [1, -1, -1, -1, 1, 1, -1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 2, 3, 4, 1, 0, 1, 1], [0, 1, 0, 0, 1, 0, -1, 1, -1, 1, 0, 1, 1, 1, 0, -1, 0, -1, 1, 1, 4, 2, 0, -1], [0, 1, 0, 1, -1, -1, 0, 1, -1, 0, 0, 1, 1, 1, 0, -2, -1, -1, 0, 0, 2, 4, 1, 0], [1, -1, 0, 1, -1, 0, 0, 0, 1, -1, -1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 4, 2], [1, -1, 0, 1, -1, 0, 0, -1, 1, 0, 0, 1, 0, 0, -1, 0, 1, 1, 1, 1, -1, 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 number216

Theta series

\[ \sum_{x \in L} q^{b(x, x)} = 1 + 216q^{2} + 191376q^{4} + O(q^{5}) \]

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)no
Number of roots216
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong A_{8}^{3}\)
Rank of \(R(L)\)24
\([L : R(L)]\)27
\(R(L)\) is primitive in \(L\)no: \(L / R(L) \cong \mathbb{Z}/3 \oplus \mathbb{Z}/9\)

Components of \(\Phi(L)\)

ComponentTypeValues of \(b(r, r)\)Number of roots\(\mathbb{Z}\Phi_c\)
\(\Phi_{1}\)\(A_{8}\)\(\{2\}\)72\(\cong A_{8}\)
\(\Phi_{2}\)\(A_{8}\)\(\{2\}\)72\(\cong A_{8}\)
\(\Phi_{3}\)\(A_{8}\)\(\{2\}\)72\(\cong A_{8}\)

Simple roots

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

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

Notes

The components \(b(e_i, e_j)\) are those of the section GRAM of the entry NIEM_3A8 of the Catalogue of Lattices.

References

  1. G. Nebe and N. J. A. Sloane, Catalogue of Lattices, entry NIEM_3A8. https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/NIEM_3A8.html