Tag 028W

\(N(A_{4}^{6})\)

N(A4^6)·NIEM_6A4·Niemeier lattice with root system N(A4^6)

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

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)\(e_{13}\)\(e_{14}\)\(e_{15}\)\(e_{16}\)\(e_{17}\)\(e_{18}\)\(e_{19}\)\(e_{20}\)\(e_{21}\)\(e_{22}\)\(e_{23}\)\(e_{24}\)
\(e_{1}\)211000000001111111111100
\(e_{2}\)121000000000001111111100
\(e_{3}\)112000000001111111111100
\(e_{4}\)00020000000100111-1-100011
\(e_{5}\)00002000000-100100001-1-111
\(e_{6}\)000002100000001-1-11-1-1-1-1-1-1
\(e_{7}\)000001200000-1-11-1-11-1-1-1-1-1-1
\(e_{8}\)000000021011-1-101111-1-1111
\(e_{9}\)000000012001-1-101111-1-1011
\(e_{10}\)00000000021111-1001-111111
\(e_{11}\)00000001012100-1001000111
\(e_{12}\)1011-10011114110221001211
\(e_{13}\)101000-1-1-1101430000022100
\(e_{14}\)101000-1-1-1101340000022100
\(e_{15}\)111111100-1-1000411000-1-100
\(e_{16}\)11110-1-111002001430111222
\(e_{17}\)11110-1-111002001340111222
\(e_{18}\)111-101111111000004100100
\(e_{19}\)111-10-1-111-100000111400100
\(e_{20}\)11101-1-1-1-1100220110042111
\(e_{21}\)1110-1-1-1-1-110122-1110024200
\(e_{22}\)1110-1-1-11011211-1221112411
\(e_{23}\)00011-1-111111000220010143
\(e_{24}\)00011-1-111111000220010134
 
[[2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0], [1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0], [1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0], [0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, -1, -1, 0, 0, 0, 1, 1], [0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 1, -1, -1, 1, 1], [0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 1, -1, -1, 1, -1, -1, -1, -1, -1, -1], [0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, -1, -1, 1, -1, -1, 1, -1, -1, -1, -1, -1, -1], [0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 1, 1, -1, -1, 0, 1, 1, 1, 1, -1, -1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 1, -1, -1, 0, 1, 1, 1, 1, -1, -1, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 1, 1, 1, -1, 0, 0, 1, -1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 1, 0, 0, -1, 0, 0, 1, 0, 0, 0, 1, 1, 1], [1, 0, 1, 1, -1, 0, 0, 1, 1, 1, 1, 4, 1, 1, 0, 2, 2, 1, 0, 0, 1, 2, 1, 1], [1, 0, 1, 0, 0, 0, -1, -1, -1, 1, 0, 1, 4, 3, 0, 0, 0, 0, 0, 2, 2, 1, 0, 0], [1, 0, 1, 0, 0, 0, -1, -1, -1, 1, 0, 1, 3, 4, 0, 0, 0, 0, 0, 2, 2, 1, 0, 0], [1, 1, 1, 1, 1, 1, 1, 0, 0, -1, -1, 0, 0, 0, 4, 1, 1, 0, 0, 0, -1, -1, 0, 0], [1, 1, 1, 1, 0, -1, -1, 1, 1, 0, 0, 2, 0, 0, 1, 4, 3, 0, 1, 1, 1, 2, 2, 2], [1, 1, 1, 1, 0, -1, -1, 1, 1, 0, 0, 2, 0, 0, 1, 3, 4, 0, 1, 1, 1, 2, 2, 2], [1, 1, 1, -1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 4, 1, 0, 0, 1, 0, 0], [1, 1, 1, -1, 0, -1, -1, 1, 1, -1, 0, 0, 0, 0, 0, 1, 1, 1, 4, 0, 0, 1, 0, 0], [1, 1, 1, 0, 1, -1, -1, -1, -1, 1, 0, 0, 2, 2, 0, 1, 1, 0, 0, 4, 2, 1, 1, 1], [1, 1, 1, 0, -1, -1, -1, -1, -1, 1, 0, 1, 2, 2, -1, 1, 1, 0, 0, 2, 4, 2, 0, 0], [1, 1, 1, 0, -1, -1, -1, 1, 0, 1, 1, 2, 1, 1, -1, 2, 2, 1, 1, 1, 2, 4, 1, 1], [0, 0, 0, 1, 1, -1, -1, 1, 1, 1, 1, 1, 0, 0, 0, 2, 2, 0, 0, 1, 0, 1, 4, 3], [0, 0, 0, 1, 1, -1, -1, 1, 1, 1, 1, 1, 0, 0, 0, 2, 2, 0, 0, 1, 0, 1, 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 number120

Theta series

\[ \sum_{x \in L} q^{b(x, x)} = 1 + 120q^{2} + 193680q^{4} + O(q^{5}) \]

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)no
Number of roots120
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong A_{4}^{6}\)
Rank of \(R(L)\)24
\([L : R(L)]\)125
\(R(L)\) is primitive in \(L\)no: \(L / R(L) \cong (\mathbb{Z}/5)^{3}\)

Components of \(\Phi(L)\)

ComponentTypeValues of \(b(r, r)\)Number of roots\(\mathbb{Z}\Phi_c\)
\(\Phi_{1}\)\(A_{4}\)\(\{2\}\)20\(\cong A_{4}\)
\(\Phi_{2}\)\(A_{4}\)\(\{2\}\)20\(\cong A_{4}\)
\(\Phi_{3}\)\(A_{4}\)\(\{2\}\)20\(\cong A_{4}\)
\(\Phi_{4}\)\(A_{4}\)\(\{2\}\)20\(\cong A_{4}\)
\(\Phi_{5}\)\(A_{4}\)\(\{2\}\)20\(\cong A_{4}\)
\(\Phi_{6}\)\(A_{4}\)\(\{2\}\)20\(\cong A_{4}\)

Simple roots

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

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

Notes

The components \(b(e_i, e_j)\) are those of the section GRAM of the entry NIEM_6A4 of the Catalogue of Lattices.

References

  1. G. Nebe and N. J. A. Sloane, Catalogue of Lattices, entry NIEM_6A4. https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/NIEM_6A4.html