Tag 029E

\(D_{16}^{+} \oplus E_{8}\)

D16+ + E8·NIEM_D16_E8·Niemeier lattice with root system D16+ + E8

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

In this basis, \(L = {}\)\(\mathbb{Z}\{e_{1}, e_{2}, e_{3}, e_{4}, e_{5}, e_{6}, e_{7}, e_{8}, e_{9}, e_{10}, e_{11}, e_{12}, e_{13}, e_{14}, e_{23}, e_{24}\}\) \(\oplus\) \(\mathbb{Z}\{e_{15}, \dots, e_{22}\}\), an orthogonal sum.

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)\(e_{13}\)\(e_{14}\)\(e_{15}\)\(e_{16}\)\(e_{17}\)\(e_{18}\)\(e_{19}\)\(e_{20}\)\(e_{21}\)\(e_{22}\)\(e_{23}\)\(e_{24}\)
\(e_{1}\)20001-10-11-1-11110000000000
\(e_{2}\)021111100000000000000010
\(e_{3}\)012111100000000000000010
\(e_{4}\)011211100000000000000000
\(e_{5}\)1111201-11-1-1111000000000-1
\(e_{6}\)-11110211-111-1-1-10000000010
\(e_{7}\)01111120000000000000000-1
\(e_{8}\)-1000-1102-111-1-1-10000000000
\(e_{9}\)10001-10-12-1-11110000000000
\(e_{10}\)-1000-1101-121-1-1-10000000011
\(e_{11}\)-1000-1101-112-1-1-10000000011
\(e_{12}\)10001-10-11-1-121100000000-10
\(e_{13}\)10001-10-11-1-112100000000-1-1
\(e_{14}\)10001-10-11-1-111200000000-1-1
\(e_{15}\)000000000000002011111100
\(e_{16}\)0000000000000002100-1-1-100
\(e_{17}\)000000000000001121100000
\(e_{18}\)000000000000001012111100
\(e_{19}\)000000000000001011201100
\(e_{20}\)000000000000001-101021100
\(e_{21}\)000000000000001-101112100
\(e_{22}\)000000000000001-101111200
\(e_{23}\)01100100011-1-1-10000000042
\(e_{24}\)0000-10-100110-1-10000000024
 
[[2, 0, 0, 0, 1, -1, 0, -1, 1, -1, -1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 2, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0], [0, 1, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0], [0, 1, 1, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 1, 1, 1, 2, 0, 1, -1, 1, -1, -1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1], [-1, 1, 1, 1, 0, 2, 1, 1, -1, 1, 1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0], [0, 1, 1, 1, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1], [-1, 0, 0, 0, -1, 1, 0, 2, -1, 1, 1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 1, -1, 0, -1, 2, -1, -1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, -1, 1, 0, 1, -1, 2, 1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [-1, 0, 0, 0, -1, 1, 0, 1, -1, 1, 2, -1, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [1, 0, 0, 0, 1, -1, 0, -1, 1, -1, -1, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0], [1, 0, 0, 0, 1, -1, 0, -1, 1, -1, -1, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1], [1, 0, 0, 0, 1, -1, 0, -1, 1, -1, -1, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 1, 1, 1, 1, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, -1, -1, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 1, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 1, 1, 1, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 0, 1, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, 0, 2, 1, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, 1, 1, 2, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, 1, 1, 1, 2, 0, 0], [0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 4, 2], [0, 0, 0, 0, -1, 0, -1, 0, 0, 1, 1, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 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 number720

Theta series

\[ \sum_{x \in L} q^{b(x, x)} = 1 + 720q^{2} + 179280q^{4} + O(q^{5}) \]

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)no
Number of roots720
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong D_{16} \oplus E_{8}\)
Rank of \(R(L)\)24
\([L : R(L)]\)2
\(R(L)\) is primitive in \(L\)no: \(L / R(L) \cong \mathbb{Z}/2\)

Components of \(\Phi(L)\)

ComponentTypeValues of \(b(r, r)\)Number of roots\(\mathbb{Z}\Phi_c\)
\(\Phi_{1}\)\(D_{16}\)\(\{2\}\)480\(\cong D_{16}\)
\(\Phi_{2}\)\(E_{8}\)\(\{2\}\)240\(\cong E_{8}\)

Simple roots

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

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

Notes

The components \(b(e_i, e_j)\) are those of the section GRAM of the entry NIEM_D16_E8 of the Catalogue of Lattices.

References

  1. G. Nebe and N. J. A. Sloane, Catalogue of Lattices, entry NIEM_D16_E8. https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/NIEM_D16_E8.html