Tag 0291

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

N(E6^4)·NIEM_4E6·Niemeier lattice with root system N(E6^4)

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

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)\(e_{13}\)\(e_{14}\)\(e_{15}\)\(e_{16}\)\(e_{17}\)\(e_{18}\)\(e_{19}\)\(e_{20}\)\(e_{21}\)\(e_{22}\)\(e_{23}\)\(e_{24}\)
\(e_{1}\)211000000000000011110000
\(e_{2}\)121100000000000010001100
\(e_{3}\)112100000000000000000000
\(e_{4}\)01120000000000000-1-1-11100
\(e_{5}\)00002100000000000100-1011
\(e_{6}\)000012000000000000000011
\(e_{7}\)0000002001100000000-110-11
\(e_{8}\)000000020-1-11000011-11-1-11-1
\(e_{9}\)00000000210-10000-1-1100100
\(e_{10}\)0000001-1121-10000-1-11-111-11
\(e_{11}\)0000001-101200000-1-11-111-11
\(e_{12}\)00000001-1-102000011-11-1-11-1
\(e_{13}\)00000000000021-1-110-1-10-11-1
\(e_{14}\)0000000000001200100-10-11-1
\(e_{15}\)000000000000-1021-101101-11
\(e_{16}\)000000000000-101200100000
\(e_{17}\)11000001-1-1-1111-1042-100-22-2
\(e_{18}\)100-11001-1-1-1100002402-2-210
\(e_{19}\)100-1000-1111-1-1011-104101-11
\(e_{20}\)100-100-110-1-11-1-1100214-2000
\(e_{21}\)0101-101-1011-100000-20-242-21
\(e_{22}\)0101000-1111-1-1-110-2-21024-22
\(e_{23}\)000011-110-1-1111-1021-10-2-24-2
\(e_{24}\)0000111-1011-1-1-110-201012-24
 
[[2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0], [1, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0], [1, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, -1, 1, 1, 0, 0], [0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 1, 1], [0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, -1, 1], [0, 0, 0, 0, 0, 0, 0, 2, 0, -1, -1, 1, 0, 0, 0, 0, 1, 1, -1, 1, -1, -1, 1, -1], [0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, -1, 0, 0, 0, 0, -1, -1, 1, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 1, -1, 1, 2, 1, -1, 0, 0, 0, 0, -1, -1, 1, -1, 1, 1, -1, 1], [0, 0, 0, 0, 0, 0, 1, -1, 0, 1, 2, 0, 0, 0, 0, 0, -1, -1, 1, -1, 1, 1, -1, 1], [0, 0, 0, 0, 0, 0, 0, 1, -1, -1, 0, 2, 0, 0, 0, 0, 1, 1, -1, 1, -1, -1, 1, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, -1, -1, 1, 0, -1, -1, 0, -1, 1, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 1, 0, 0, -1, 0, -1, 1, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 2, 1, -1, 0, 1, 1, 0, 1, -1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1, 2, 0, 0, 1, 0, 0, 0, 0, 0], [1, 1, 0, 0, 0, 0, 0, 1, -1, -1, -1, 1, 1, 1, -1, 0, 4, 2, -1, 0, 0, -2, 2, -2], [1, 0, 0, -1, 1, 0, 0, 1, -1, -1, -1, 1, 0, 0, 0, 0, 2, 4, 0, 2, -2, -2, 1, 0], [1, 0, 0, -1, 0, 0, 0, -1, 1, 1, 1, -1, -1, 0, 1, 1, -1, 0, 4, 1, 0, 1, -1, 1], [1, 0, 0, -1, 0, 0, -1, 1, 0, -1, -1, 1, -1, -1, 1, 0, 0, 2, 1, 4, -2, 0, 0, 0], [0, 1, 0, 1, -1, 0, 1, -1, 0, 1, 1, -1, 0, 0, 0, 0, 0, -2, 0, -2, 4, 2, -2, 1], [0, 1, 0, 1, 0, 0, 0, -1, 1, 1, 1, -1, -1, -1, 1, 0, -2, -2, 1, 0, 2, 4, -2, 2], [0, 0, 0, 0, 1, 1, -1, 1, 0, -1, -1, 1, 1, 1, -1, 0, 2, 1, -1, 0, -2, -2, 4, -2], [0, 0, 0, 0, 1, 1, 1, -1, 0, 1, 1, -1, -1, -1, 1, 0, -2, 0, 1, 0, 1, 2, -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 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
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong E_{6}^{4}\)
Rank of \(R(L)\)24
\([L : R(L)]\)9
\(R(L)\) is primitive in \(L\)no: \(L / R(L) \cong (\mathbb{Z}/3)^{2}\)

Components of \(\Phi(L)\)

ComponentTypeValues of \(b(r, r)\)Number of roots\(\mathbb{Z}\Phi_c\)
\(\Phi_{1}\)\(E_{6}\)\(\{2\}\)72\(\cong E_{6}\)
\(\Phi_{2}\)\(E_{6}\)\(\{2\}\)72\(\cong E_{6}\)
\(\Phi_{3}\)\(E_{6}\)\(\{2\}\)72\(\cong E_{6}\)
\(\Phi_{4}\)\(E_{6}\)\(\{2\}\)72\(\cong E_{6}\)

Simple roots

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

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

Notes

The components \(b(e_i, e_j)\) are those of the section GRAM of the entry NIEM_4E6 of the Catalogue of Lattices.

References

  1. G. Nebe and N. J. A. Sloane, Catalogue of Lattices, entry NIEM_4E6. https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/NIEM_4E6.html