Tag 0299

\(N(A_{11} D_{7} E_{6})\)

N(A11 D7 E6)·NIEM_A11_D7_E6·Niemeier lattice with root system N(A11 D7 E6)

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

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)\(e_{13}\)\(e_{14}\)\(e_{15}\)\(e_{16}\)\(e_{17}\)\(e_{18}\)\(e_{19}\)\(e_{20}\)\(e_{21}\)\(e_{22}\)\(e_{23}\)\(e_{24}\)
\(e_{1}\)2-1-1-1-1-1-100000000101100011
\(e_{2}\)-121111100000000-11-1-1111-1-1
\(e_{3}\)-11211110000000000-1-1000-1-1
\(e_{4}\)-1112111000000000000000-1-1
\(e_{5}\)-11112110000000000-1-1000-1-1
\(e_{6}\)-11111210000000000-1-1000-1-1
\(e_{7}\)-111111200000000-11-1-1000-1-1
\(e_{8}\)00000002-1-1-1-1000-10-1111-111
\(e_{9}\)0000000-1211110000100-10-1-1
\(e_{10}\)0000000-1121110000100-10-1-1
\(e_{11}\)0000000-111210-101-11-1-1-11-1-1
\(e_{12}\)0000000-11112000100-100100
\(e_{13}\)000000001100200-11111-1-1-1-1
\(e_{14}\)0000000000-10020-11-1111-111
\(e_{15}\)00000000000000200000-1110
\(e_{16}\)1-10000-1-10011-1-104-21-1-2-1100
\(e_{17}\)0100001000-10110-24-1021000
\(e_{18}\)1-1-10-1-1-1-111101-101-141-1-20-1-1
\(e_{19}\)1-1-10-1-1-1100-1-1110-101410-211
\(e_{20}\)0100000100-10110-22-1142000
\(e_{21}\)01000001-1-1-10-11-1-11-2024012
\(e_{22}\)0100000-10011-1-11100-20040-1
\(e_{23}\)1-1-1-1-1-1-11-1-1-10-11100-1101043
\(e_{24}\)1-1-1-1-1-1-11-1-1-10-11000-1102-134
 
[[2, -1, -1, -1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1], [-1, 2, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, -1, -1, 1, 1, 1, -1, -1], [-1, 1, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, -1, -1], [-1, 1, 1, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1], [-1, 1, 1, 1, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, -1, -1], [-1, 1, 1, 1, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, -1, -1], [-1, 1, 1, 1, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, -1, -1, 0, 0, 0, -1, -1], [0, 0, 0, 0, 0, 0, 0, 2, -1, -1, -1, -1, 0, 0, 0, -1, 0, -1, 1, 1, 1, -1, 1, 1], [0, 0, 0, 0, 0, 0, 0, -1, 2, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, -1, 0, -1, -1], [0, 0, 0, 0, 0, 0, 0, -1, 1, 2, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, -1, 0, -1, -1], [0, 0, 0, 0, 0, 0, 0, -1, 1, 1, 2, 1, 0, -1, 0, 1, -1, 1, -1, -1, -1, 1, -1, -1], [0, 0, 0, 0, 0, 0, 0, -1, 1, 1, 1, 2, 0, 0, 0, 1, 0, 0, -1, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, -1, 1, 1, 1, 1, -1, -1, -1, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 2, 0, -1, 1, -1, 1, 1, 1, -1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, -1, 1, 1, 0], [1, -1, 0, 0, 0, 0, -1, -1, 0, 0, 1, 1, -1, -1, 0, 4, -2, 1, -1, -2, -1, 1, 0, 0], [0, 1, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0, 1, 1, 0, -2, 4, -1, 0, 2, 1, 0, 0, 0], [1, -1, -1, 0, -1, -1, -1, -1, 1, 1, 1, 0, 1, -1, 0, 1, -1, 4, 1, -1, -2, 0, -1, -1], [1, -1, -1, 0, -1, -1, -1, 1, 0, 0, -1, -1, 1, 1, 0, -1, 0, 1, 4, 1, 0, -2, 1, 1], [0, 1, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 1, 1, 0, -2, 2, -1, 1, 4, 2, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 1, -1, -1, -1, 0, -1, 1, -1, -1, 1, -2, 0, 2, 4, 0, 1, 2], [0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 1, 1, -1, -1, 1, 1, 0, 0, -2, 0, 0, 4, 0, -1], [1, -1, -1, -1, -1, -1, -1, 1, -1, -1, -1, 0, -1, 1, 1, 0, 0, -1, 1, 0, 1, 0, 4, 3], [1, -1, -1, -1, -1, -1, -1, 1, -1, -1, -1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 2, -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 number288

Theta series

\[ \sum_{x \in L} q^{b(x, x)} = 1 + 288q^{2} + 189648q^{4} + O(q^{5}) \]

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)no
Number of roots288
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong A_{11} \oplus D_{7} \oplus E_{6}\)
Rank of \(R(L)\)24
\([L : R(L)]\)12
\(R(L)\) is primitive in \(L\)no: \(L / R(L) \cong \mathbb{Z}/12\)

Components of \(\Phi(L)\)

ComponentTypeValues of \(b(r, r)\)Number of roots\(\mathbb{Z}\Phi_c\)
\(\Phi_{1}\)\(A_{11}\)\(\{2\}\)132\(\cong A_{11}\)
\(\Phi_{2}\)\(D_{7}\)\(\{2\}\)84\(\cong D_{7}\)
\(\Phi_{3}\)\(E_{6}\)\(\{2\}\)72\(\cong E_{6}\)

Simple roots

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

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

Notes

The components \(b(e_i, e_j)\) are those of the section GRAM of the entry NIEM_A11_D7_E6 of the Catalogue of Lattices.

References

  1. G. Nebe and N. J. A. Sloane, Catalogue of Lattices, entry NIEM_A11_D7_E6. https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/NIEM_A11_D7_E6.html