Tag 0295

\(N(A_{7}^{2} D_{5}^{2})\)

N(A7^2 D5^2)·NIEM_2A7_2D5·Niemeier lattice with root system N(A7^2 D5^2)

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

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)\(e_{13}\)\(e_{14}\)\(e_{15}\)\(e_{16}\)\(e_{17}\)\(e_{18}\)\(e_{19}\)\(e_{20}\)\(e_{21}\)\(e_{22}\)\(e_{23}\)\(e_{24}\)
\(e_{1}\)2-10000000000101-100000011
\(e_{2}\)-12-1000000000-1-1-11111111-1-1
\(e_{3}\)0-12000000000100-1-1-1-1-1-1-100
\(e_{4}\)000211-1000001-11111-100-111
\(e_{5}\)000121-1000001-11111-1-10011
\(e_{6}\)000112-1000001-10011-1-11011
\(e_{7}\)000-1-1-1200000-11-1-1-1-111-11-1-1
\(e_{8}\)000000020110-10-1-111100000
\(e_{9}\)0000000021-100-11100-111-11-1
\(e_{10}\)000000011200-1-1001101100-1
\(e_{11}\)00000001-1021-10-1-1111001-10
\(e_{12}\)000000000012-1-100110111-1-1
\(e_{13}\)1-11111-1-10-1-1-14020-1-1-2-2-1-222
\(e_{14}\)0-10-1-1-110-1-10-104-1-1-2-21-1-2001
\(e_{15}\)1-10110-1-110-102-14200-200-221
\(e_{16}\)-11-1110-1-110-100-12411-111-110
\(e_{17}\)01-1111-110111-1-20143112100
\(e_{18}\)01-1111-110111-1-20134112100
\(e_{19}\)01-1-1-1-111-1010-21-2-1114102-10
\(e_{20}\)01-10-1-1101101-2-101111411-1-2
\(e_{21}\)01-1001-101101-1-2012201410-1
\(e_{22}\)01-1-10010-1011-20-2-1112114-2-1
\(e_{23}\)1-10111-1010-1-1202100-1-10-242
\(e_{24}\)1-10111-10-1-10-12110000-2-1-124
 
[[2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, -1, 0, 0, 0, 0, 0, 0, 1, 1], [-1, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, -1, 1, 1, 1, 1, 1, 1, 1, -1, -1], [0, -1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1, -1, -1, -1, -1, -1, -1, 0, 0], [0, 0, 0, 2, 1, 1, -1, 0, 0, 0, 0, 0, 1, -1, 1, 1, 1, 1, -1, 0, 0, -1, 1, 1], [0, 0, 0, 1, 2, 1, -1, 0, 0, 0, 0, 0, 1, -1, 1, 1, 1, 1, -1, -1, 0, 0, 1, 1], [0, 0, 0, 1, 1, 2, -1, 0, 0, 0, 0, 0, 1, -1, 0, 0, 1, 1, -1, -1, 1, 0, 1, 1], [0, 0, 0, -1, -1, -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, 0, 1, 1, 0, -1, 0, -1, -1, 1, 1, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, 1, -1, 0, 0, -1, 1, 1, 0, 0, -1, 1, 1, -1, 1, -1], [0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 0, 0, -1, -1, 0, 0, 1, 1, 0, 1, 1, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 2, 1, -1, 0, -1, -1, 1, 1, 1, 0, 0, 1, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, -1, -1, 0, 0, 1, 1, 0, 1, 1, 1, -1, -1], [1, -1, 1, 1, 1, 1, -1, -1, 0, -1, -1, -1, 4, 0, 2, 0, -1, -1, -2, -2, -1, -2, 2, 2], [0, -1, 0, -1, -1, -1, 1, 0, -1, -1, 0, -1, 0, 4, -1, -1, -2, -2, 1, -1, -2, 0, 0, 1], [1, -1, 0, 1, 1, 0, -1, -1, 1, 0, -1, 0, 2, -1, 4, 2, 0, 0, -2, 0, 0, -2, 2, 1], [-1, 1, -1, 1, 1, 0, -1, -1, 1, 0, -1, 0, 0, -1, 2, 4, 1, 1, -1, 1, 1, -1, 1, 0], [0, 1, -1, 1, 1, 1, -1, 1, 0, 1, 1, 1, -1, -2, 0, 1, 4, 3, 1, 1, 2, 1, 0, 0], [0, 1, -1, 1, 1, 1, -1, 1, 0, 1, 1, 1, -1, -2, 0, 1, 3, 4, 1, 1, 2, 1, 0, 0], [0, 1, -1, -1, -1, -1, 1, 1, -1, 0, 1, 0, -2, 1, -2, -1, 1, 1, 4, 1, 0, 2, -1, 0], [0, 1, -1, 0, -1, -1, 1, 0, 1, 1, 0, 1, -2, -1, 0, 1, 1, 1, 1, 4, 1, 1, -1, -2], [0, 1, -1, 0, 0, 1, -1, 0, 1, 1, 0, 1, -1, -2, 0, 1, 2, 2, 0, 1, 4, 1, 0, -1], [0, 1, -1, -1, 0, 0, 1, 0, -1, 0, 1, 1, -2, 0, -2, -1, 1, 1, 2, 1, 1, 4, -2, -1], [1, -1, 0, 1, 1, 1, -1, 0, 1, 0, -1, -1, 2, 0, 2, 1, 0, 0, -1, -1, 0, -2, 4, 2], [1, -1, 0, 1, 1, 1, -1, 0, -1, -1, 0, -1, 2, 1, 1, 0, 0, 0, 0, -2, -1, -1, 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 number192

Theta series

\[ \sum_{x \in L} q^{b(x, x)} = 1 + 192q^{2} + 191952q^{4} + O(q^{5}) \]

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)no
Number of roots192
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong A_{7}^{2} \oplus D_{5}^{2}\)
Rank of \(R(L)\)24
\([L : R(L)]\)32
\(R(L)\) is primitive in \(L\)no: \(L / R(L) \cong \mathbb{Z}/4 \oplus \mathbb{Z}/8\)

Components of \(\Phi(L)\)

ComponentTypeValues of \(b(r, r)\)Number of roots\(\mathbb{Z}\Phi_c\)
\(\Phi_{1}\)\(A_{7}\)\(\{2\}\)56\(\cong A_{7}\)
\(\Phi_{2}\)\(A_{7}\)\(\{2\}\)56\(\cong A_{7}\)
\(\Phi_{3}\)\(D_{5}\)\(\{2\}\)40\(\cong D_{5}\)
\(\Phi_{4}\)\(D_{5}\)\(\{2\}\)40\(\cong D_{5}\)

Simple roots

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

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

Notes

The components \(b(e_i, e_j)\) are those of the section GRAM of the entry NIEM_2A7_2D5 of the Catalogue of Lattices.

References

  1. G. Nebe and N. J. A. Sloane, Catalogue of Lattices, entry NIEM_2A7_2D5. https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/NIEM_2A7_2D5.html