Tag 028Y

\(N(A_{5}^{4} D_{4})\)

N(A5^4 D4)·NIEM_4A5_D4·Niemeier lattice with root system N(A5^4 D4)

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

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)\(e_{13}\)\(e_{14}\)\(e_{15}\)\(e_{16}\)\(e_{17}\)\(e_{18}\)\(e_{19}\)\(e_{20}\)\(e_{21}\)\(e_{22}\)\(e_{23}\)\(e_{24}\)
\(e_{1}\)2-10000000000-11011-1010000
\(e_{2}\)-12-10000000001-11-1-111-10000
\(e_{3}\)0-1200000000001-101-1-100000
\(e_{4}\)0002000000000111-101-11111
\(e_{5}\)00002000000001011-1110111
\(e_{6}\)0000020000000-1-1001-1-1-1-1-1-1
\(e_{7}\)000000200000-10000001-1000
\(e_{8}\)00000002-1-1-10100111-1-1-1-1-1-1
\(e_{9}\)0000000-1211-100100-1110011
\(e_{10}\)0000000-11210-100-1-1-1111000
\(e_{11}\)0000000-11120-101-1-1-1111111
\(e_{12}\)00000000-1002-10-1-1-110011-1-1
\(e_{13}\)-110000-110-1-1-14011110-2-1-100
\(e_{14}\)1-1111-100000004121-2111212
\(e_{15}\)01-110-100101-11141-1-1201122
\(e_{16}\)1-10110010-1-1-112142-100-1011
\(e_{17}\)1-11-110010-1-1-111-124-1-11-2-100
\(e_{18}\)-11-10-1101-1-1-111-2-1-1-14-1-2-1-1-2-2
\(e_{19}\)01-111-10-111100120-1-1412222
\(e_{20}\)1-10-11-11-11110-21001-2140111
\(e_{21}\)00010-1-1-10111-111-1-2-1204211
\(e_{22}\)00011-10-10011-1210-1-1212422
\(e_{23}\)00011-10-1101-101210-2211243
\(e_{24}\)00011-10-1101-102210-2211234
 
[[2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 1, 1, -1, 0, 1, 0, 0, 0, 0], [-1, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 1, -1, -1, 1, 1, -1, 0, 0, 0, 0], [0, -1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, -1, 0, 0, 0, 0, 0], [0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, -1, 0, 1, -1, 1, 1, 1, 1], [0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, -1, 1, 1, 0, 1, 1, 1], [0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0, 0, 1, -1, -1, -1, -1, -1, -1], [0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 2, -1, -1, -1, 0, 1, 0, 0, 1, 1, 1, -1, -1, -1, -1, -1, -1], [0, 0, 0, 0, 0, 0, 0, -1, 2, 1, 1, -1, 0, 0, 1, 0, 0, -1, 1, 1, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, -1, 1, 2, 1, 0, -1, 0, 0, -1, -1, -1, 1, 1, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 1, 1, 2, 0, -1, 0, 1, -1, -1, -1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 2, -1, 0, -1, -1, -1, 1, 0, 0, 1, 1, -1, -1], [-1, 1, 0, 0, 0, 0, -1, 1, 0, -1, -1, -1, 4, 0, 1, 1, 1, 1, 0, -2, -1, -1, 0, 0], [1, -1, 1, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, 4, 1, 2, 1, -2, 1, 1, 1, 2, 1, 2], [0, 1, -1, 1, 0, -1, 0, 0, 1, 0, 1, -1, 1, 1, 4, 1, -1, -1, 2, 0, 1, 1, 2, 2], [1, -1, 0, 1, 1, 0, 0, 1, 0, -1, -1, -1, 1, 2, 1, 4, 2, -1, 0, 0, -1, 0, 1, 1], [1, -1, 1, -1, 1, 0, 0, 1, 0, -1, -1, -1, 1, 1, -1, 2, 4, -1, -1, 1, -2, -1, 0, 0], [-1, 1, -1, 0, -1, 1, 0, 1, -1, -1, -1, 1, 1, -2, -1, -1, -1, 4, -1, -2, -1, -1, -2, -2], [0, 1, -1, 1, 1, -1, 0, -1, 1, 1, 1, 0, 0, 1, 2, 0, -1, -1, 4, 1, 2, 2, 2, 2], [1, -1, 0, -1, 1, -1, 1, -1, 1, 1, 1, 0, -2, 1, 0, 0, 1, -2, 1, 4, 0, 1, 1, 1], [0, 0, 0, 1, 0, -1, -1, -1, 0, 1, 1, 1, -1, 1, 1, -1, -2, -1, 2, 0, 4, 2, 1, 1], [0, 0, 0, 1, 1, -1, 0, -1, 0, 0, 1, 1, -1, 2, 1, 0, -1, -1, 2, 1, 2, 4, 2, 2], [0, 0, 0, 1, 1, -1, 0, -1, 1, 0, 1, -1, 0, 1, 2, 1, 0, -2, 2, 1, 1, 2, 4, 3], [0, 0, 0, 1, 1, -1, 0, -1, 1, 0, 1, -1, 0, 2, 2, 1, 0, -2, 2, 1, 1, 2, 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 number144

Theta series

\[ \sum_{x \in L} q^{b(x, x)} = 1 + 144q^{2} + 193104q^{4} + O(q^{5}) \]

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)no
Number of roots144
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong A_{5}^{4} \oplus D_{4}\)
Rank of \(R(L)\)24
\([L : R(L)]\)72
\(R(L)\) is primitive in \(L\)no: \(L / R(L) \cong \mathbb{Z}/2 \oplus (\mathbb{Z}/6)^{2}\)

Components of \(\Phi(L)\)

ComponentTypeValues of \(b(r, r)\)Number of roots\(\mathbb{Z}\Phi_c\)
\(\Phi_{1}\)\(A_{5}\)\(\{2\}\)30\(\cong A_{5}\)
\(\Phi_{2}\)\(A_{5}\)\(\{2\}\)30\(\cong A_{5}\)
\(\Phi_{3}\)\(A_{5}\)\(\{2\}\)30\(\cong A_{5}\)
\(\Phi_{4}\)\(A_{5}\)\(\{2\}\)30\(\cong A_{5}\)
\(\Phi_{5}\)\(D_{4}\)\(\{2\}\)24\(\cong D_{4}\)

Simple roots

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

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

Notes

The components \(b(e_i, e_j)\) are those of the section GRAM of the entry NIEM_4A5_D4 of the Catalogue of Lattices.

References

  1. G. Nebe and N. J. A. Sloane, Catalogue of Lattices, entry NIEM_4A5_D4. https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/NIEM_4A5_D4.html