Tag 028V

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

N(A3^8)·NIEM_8A3·Niemeier lattice with root system N(A3^8)

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

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)\(e_{13}\)\(e_{14}\)\(e_{15}\)\(e_{16}\)\(e_{17}\)\(e_{18}\)\(e_{19}\)\(e_{20}\)\(e_{21}\)\(e_{22}\)\(e_{23}\)\(e_{24}\)
\(e_{1}\)21-1000000000001001111000
\(e_{2}\)12-10000000000-10001111000
\(e_{3}\)-1-1200000000001-111-1-1-1-1000
\(e_{4}\)00020000000000-1-101-100110
\(e_{5}\)0000200000000000-10000001
\(e_{6}\)0000020000000100-1-1111-1-1-1
\(e_{7}\)00000020000000100101-1001
\(e_{8}\)00000002-100001-1000001-1-1-1
\(e_{9}\)0000000-12-1000-11110-11-1111
\(e_{10}\)00000000-1200010-1-101-10-1-1-1
\(e_{11}\)000000000021-1-11-1-1-11-11-1-11
\(e_{12}\)0000000000120-110-1-1001-1-11
\(e_{13}\)0000000000-10210001-110000
\(e_{14}\)0-1100101-11-1-114-1000000-1-1-2
\(e_{15}\)10-1-1001-110110-140-10110-1-12
\(e_{16}\)001-100001-1-1000042-1-11-1000
\(e_{17}\)0010-1-1001-1-1-100-1241-20-2220
\(e_{18}\)11-110-11000-1-1100-114-11-1221
\(e_{19}\)11-1-10100-1110-101-1-2-1402-2-2-1
\(e_{20}\)11-1001101-1-10101101040000
\(e_{21}\)11-1001-11-1011000-1-2-1204-2-2-1
\(e_{22}\)00010-10-11-1-1-10-1-1022-20-2431
\(e_{23}\)00010-10-11-1-1-10-1-1022-20-2341
\(e_{24}\)00001-11-11-1110-22001-10-1114
 
[[2, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0], [1, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0], [-1, -1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 1, 1, -1, -1, -1, -1, 0, 0, 0], [0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0, 1, -1, 0, 0, 1, 1, 0], [0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1, -1, 1, 1, 1, -1, -1, -1], [0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, -1, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 2, -1, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 1, -1, -1, -1], [0, 0, 0, 0, 0, 0, 0, -1, 2, -1, 0, 0, 0, -1, 1, 1, 1, 0, -1, 1, -1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, -1, 2, 0, 0, 0, 1, 0, -1, -1, 0, 1, -1, 0, -1, -1, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, -1, -1, 1, -1, -1, -1, 1, -1, 1, -1, -1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, -1, 1, 0, -1, -1, 0, 0, 1, -1, -1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 2, 1, 0, 0, 0, 1, -1, 1, 0, 0, 0, 0], [0, -1, 1, 0, 0, 1, 0, 1, -1, 1, -1, -1, 1, 4, -1, 0, 0, 0, 0, 0, 0, -1, -1, -2], [1, 0, -1, -1, 0, 0, 1, -1, 1, 0, 1, 1, 0, -1, 4, 0, -1, 0, 1, 1, 0, -1, -1, 2], [0, 0, 1, -1, 0, 0, 0, 0, 1, -1, -1, 0, 0, 0, 0, 4, 2, -1, -1, 1, -1, 0, 0, 0], [0, 0, 1, 0, -1, -1, 0, 0, 1, -1, -1, -1, 0, 0, -1, 2, 4, 1, -2, 0, -2, 2, 2, 0], [1, 1, -1, 1, 0, -1, 1, 0, 0, 0, -1, -1, 1, 0, 0, -1, 1, 4, -1, 1, -1, 2, 2, 1], [1, 1, -1, -1, 0, 1, 0, 0, -1, 1, 1, 0, -1, 0, 1, -1, -2, -1, 4, 0, 2, -2, -2, -1], [1, 1, -1, 0, 0, 1, 1, 0, 1, -1, -1, 0, 1, 0, 1, 1, 0, 1, 0, 4, 0, 0, 0, 0], [1, 1, -1, 0, 0, 1, -1, 1, -1, 0, 1, 1, 0, 0, 0, -1, -2, -1, 2, 0, 4, -2, -2, -1], [0, 0, 0, 1, 0, -1, 0, -1, 1, -1, -1, -1, 0, -1, -1, 0, 2, 2, -2, 0, -2, 4, 3, 1], [0, 0, 0, 1, 0, -1, 0, -1, 1, -1, -1, -1, 0, -1, -1, 0, 2, 2, -2, 0, -2, 3, 4, 1], [0, 0, 0, 0, 1, -1, 1, -1, 1, -1, 1, 1, 0, -2, 2, 0, 0, 1, -1, 0, -1, 1, 1, 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 number96

Theta series

\[ \sum_{x \in L} q^{b(x, x)} = 1 + 96q^{2} + 194256q^{4} + O(q^{5}) \]

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)no
Number of roots96
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong A_{3}^{8}\)
Rank of \(R(L)\)24
\([L : R(L)]\)256
\(R(L)\) is primitive in \(L\)no: \(L / R(L) \cong (\mathbb{Z}/4)^{4}\)

Components of \(\Phi(L)\)

ComponentTypeValues of \(b(r, r)\)Number of roots\(\mathbb{Z}\Phi_c\)
\(\Phi_{1}\)\(A_{3}\)\(\{2\}\)12\(\cong A_{3}\)
\(\Phi_{2}\)\(A_{3}\)\(\{2\}\)12\(\cong A_{3}\)
\(\Phi_{3}\)\(A_{3}\)\(\{2\}\)12\(\cong A_{3}\)
\(\Phi_{4}\)\(A_{3}\)\(\{2\}\)12\(\cong A_{3}\)
\(\Phi_{5}\)\(A_{3}\)\(\{2\}\)12\(\cong A_{3}\)
\(\Phi_{6}\)\(A_{3}\)\(\{2\}\)12\(\cong A_{3}\)
\(\Phi_{7}\)\(A_{3}\)\(\{2\}\)12\(\cong A_{3}\)
\(\Phi_{8}\)\(A_{3}\)\(\{2\}\)12\(\cong A_{3}\)

Simple roots

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

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

Notes

The components \(b(e_i, e_j)\) are those of the section GRAM of the entry NIEM_8A3 of the Catalogue of Lattices.

References

  1. G. Nebe and N. J. A. Sloane, Catalogue of Lattices, entry NIEM_8A3. https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/NIEM_8A3.html