Tag 029B

\(N(A_{17} E_{7})\)

N(A17 E7)·NIEM_A17_E7·Niemeier lattice with root system N(A17 E7)

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

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)\(e_{13}\)\(e_{14}\)\(e_{15}\)\(e_{16}\)\(e_{17}\)\(e_{18}\)\(e_{19}\)\(e_{20}\)\(e_{21}\)\(e_{22}\)\(e_{23}\)\(e_{24}\)
\(e_{1}\)211000000000000000111111
\(e_{2}\)121000000000000000111111
\(e_{3}\)112101011-1-10000000001100
\(e_{4}\)001201011-1-10000000-1-101-1-1
\(e_{5}\)000020000001000000-1-10-1-1-1
\(e_{6}\)001102011-1-10000000-1-101-1-1
\(e_{7}\)00000020010-1-10000000-1000
\(e_{8}\)001101021-1-10000000-1-100-1-1
\(e_{9}\)001101012-1-10000000-1-100-1-1
\(e_{10}\)00-1-10-11-1-121-1-10000011-1011
\(e_{11}\)00-1-10-10-1-11200-1000011-1-111
\(e_{12}\)000010-100-102100000-1-11-1-1-1
\(e_{13}\)000000-100-101200000001-1-1-1
\(e_{14}\)0000000000-100200000011-1-1
\(e_{15}\)000000000000002101-1-1-1000
\(e_{16}\)00000000000000121000-1000
\(e_{17}\)00000000000000012-1100000
\(e_{18}\)0000000000000010-12-1-1-1000
\(e_{19}\)110-1-1-10-1-111-100-101-1431122
\(e_{20}\)110-1-1-10-1-111-100-100-1341122
\(e_{21}\)111000-100-1-1111-1-10-1114100
\(e_{22}\)1111-110000-1-1-110000111411
\(e_{23}\)110-1-1-10-1-111-1-1-10000220143
\(e_{24}\)110-1-1-10-1-111-1-1-10000220134
 
[[2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1], [1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1], [1, 1, 2, 1, 0, 1, 0, 1, 1, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0], [0, 0, 1, 2, 0, 1, 0, 1, 1, -1, -1, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0, 1, -1, -1], [0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, -1, 0, -1, -1, -1], [0, 0, 1, 1, 0, 2, 0, 1, 1, -1, -1, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0, 1, -1, -1], [0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0], [0, 0, 1, 1, 0, 1, 0, 2, 1, -1, -1, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0, 0, -1, -1], [0, 0, 1, 1, 0, 1, 0, 1, 2, -1, -1, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0, 0, -1, -1], [0, 0, -1, -1, 0, -1, 1, -1, -1, 2, 1, -1, -1, 0, 0, 0, 0, 0, 1, 1, -1, 0, 1, 1], [0, 0, -1, -1, 0, -1, 0, -1, -1, 1, 2, 0, 0, -1, 0, 0, 0, 0, 1, 1, -1, -1, 1, 1], [0, 0, 0, 0, 1, 0, -1, 0, 0, -1, 0, 2, 1, 0, 0, 0, 0, 0, -1, -1, 1, -1, -1, -1], [0, 0, 0, 0, 0, 0, -1, 0, 0, -1, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 1, -1, -1, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 1, -1, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 1, -1, -1, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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, 0, 0, 0], [1, 1, 0, -1, -1, -1, 0, -1, -1, 1, 1, -1, 0, 0, -1, 0, 1, -1, 4, 3, 1, 1, 2, 2], [1, 1, 0, -1, -1, -1, 0, -1, -1, 1, 1, -1, 0, 0, -1, 0, 0, -1, 3, 4, 1, 1, 2, 2], [1, 1, 1, 0, 0, 0, -1, 0, 0, -1, -1, 1, 1, 1, -1, -1, 0, -1, 1, 1, 4, 1, 0, 0], [1, 1, 1, 1, -1, 1, 0, 0, 0, 0, -1, -1, -1, 1, 0, 0, 0, 0, 1, 1, 1, 4, 1, 1], [1, 1, 0, -1, -1, -1, 0, -1, -1, 1, 1, -1, -1, -1, 0, 0, 0, 0, 2, 2, 0, 1, 4, 3], [1, 1, 0, -1, -1, -1, 0, -1, -1, 1, 1, -1, -1, -1, 0, 0, 0, 0, 2, 2, 0, 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 number432

Theta series

\[ \sum_{x \in L} q^{b(x, x)} = 1 + 432q^{2} + 186192q^{4} + O(q^{5}) \]

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)no
Number of roots432
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong A_{17} \oplus E_{7}\)
Rank of \(R(L)\)24
\([L : R(L)]\)6
\(R(L)\) is primitive in \(L\)no: \(L / R(L) \cong \mathbb{Z}/6\)

Components of \(\Phi(L)\)

ComponentTypeValues of \(b(r, r)\)Number of roots\(\mathbb{Z}\Phi_c\)
\(\Phi_{1}\)\(A_{17}\)\(\{2\}\)306\(\cong A_{17}\)
\(\Phi_{2}\)\(E_{7}\)\(\{2\}\)126\(\cong E_{7}\)

Simple roots

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

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

Notes

The components \(b(e_i, e_j)\) are those of the section GRAM of the entry NIEM_A17_E7 of the Catalogue of Lattices.

References

  1. G. Nebe and N. J. A. Sloane, Catalogue of Lattices, entry NIEM_A17_E7. https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/NIEM_A17_E7.html