Tag 0297

\(N(A_{12}^{2})\)

N(A12^2)·NIEM_2A12·Niemeier lattice with root system N(A12^2)

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

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)\(e_{13}\)\(e_{14}\)\(e_{15}\)\(e_{16}\)\(e_{17}\)\(e_{18}\)\(e_{19}\)\(e_{20}\)\(e_{21}\)\(e_{22}\)\(e_{23}\)\(e_{24}\)
\(e_{1}\)21-1-1-1000000000000-1111111
\(e_{2}\)12-1-1-10000000000000111111
\(e_{3}\)-1-1211-1-1-100000000-100000-1-1
\(e_{4}\)-1-11210000000000000-10-1-1-1-1
\(e_{5}\)-1-11120000000000001-1-1-1-1-1-1
\(e_{6}\)00-1002110000000011-1-1-1-100
\(e_{7}\)00-1001210000000011-1-1-1-100
\(e_{8}\)00-1001120000000010-1-1-1-100
\(e_{9}\)0000000021-1-1-1-1-1-10-1000010
\(e_{10}\)0000000012000000-10-1-1111-1
\(e_{11}\)00000000-10211111-11-1-111-1-1
\(e_{12}\)00000000-10121111-11-1-11100
\(e_{13}\)00000000-10112111-11-1-111-1-1
\(e_{14}\)00000000-10111211-11-1-111-1-1
\(e_{15}\)00000000-1011112101-1-10100
\(e_{16}\)00000000-10111112-11-1-111-1-1
\(e_{17}\)00-1001110-1-1-1-1-10-14011-3-212
\(e_{18}\)-10001110-1011111104-2-200-1-1
\(e_{19}\)110-1-1-1-1-10-1-1-1-1-1-1-11-2430012
\(e_{20}\)1100-1-1-1-10-1-1-1-1-1-1-11-2340012
\(e_{21}\)110-1-1-1-1-101111101-3000430-1
\(e_{22}\)110-1-1-1-1-101111111-20003410
\(e_{23}\)11-1-1-100011-10-1-10-11-1110142
\(e_{24}\)11-1-1-10000-1-10-1-10-12-122-1024
 
[[2, 1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 1, 1, 1, 1, 1], [1, 2, -1, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1], [-1, -1, 2, 1, 1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, -1], [-1, -1, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, -1, -1, -1], [-1, -1, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, -1, -1, -1, -1, -1], [0, 0, -1, 0, 0, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, -1, -1, -1, -1, 0, 0], [0, 0, -1, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, -1, -1, -1, -1, 0, 0], [0, 0, -1, 0, 0, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, -1, -1, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, 1, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, -1, 0, -1, -1, 1, 1, 1, -1], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 2, 1, 1, 1, 1, 1, -1, 1, -1, -1, 1, 1, -1, -1], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1, 2, 1, 1, 1, 1, -1, 1, -1, -1, 1, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1, 1, 2, 1, 1, 1, -1, 1, -1, -1, 1, 1, -1, -1], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1, 1, 1, 2, 1, 1, -1, 1, -1, -1, 1, 1, -1, -1], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1, 1, 1, 1, 2, 1, 0, 1, -1, -1, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1, 1, 1, 1, 1, 2, -1, 1, -1, -1, 1, 1, -1, -1], [0, 0, -1, 0, 0, 1, 1, 1, 0, -1, -1, -1, -1, -1, 0, -1, 4, 0, 1, 1, -3, -2, 1, 2], [-1, 0, 0, 0, 1, 1, 1, 0, -1, 0, 1, 1, 1, 1, 1, 1, 0, 4, -2, -2, 0, 0, -1, -1], [1, 1, 0, -1, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, -1, 1, -2, 4, 3, 0, 0, 1, 2], [1, 1, 0, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, -1, 1, -2, 3, 4, 0, 0, 1, 2], [1, 1, 0, -1, -1, -1, -1, -1, 0, 1, 1, 1, 1, 1, 0, 1, -3, 0, 0, 0, 4, 3, 0, -1], [1, 1, 0, -1, -1, -1, -1, -1, 0, 1, 1, 1, 1, 1, 1, 1, -2, 0, 0, 0, 3, 4, 1, 0], [1, 1, -1, -1, -1, 0, 0, 0, 1, 1, -1, 0, -1, -1, 0, -1, 1, -1, 1, 1, 0, 1, 4, 2], [1, 1, -1, -1, -1, 0, 0, 0, 0, -1, -1, 0, -1, -1, 0, -1, 2, -1, 2, 2, -1, 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 number312

Theta series

\[ \sum_{x \in L} q^{b(x, x)} = 1 + 312q^{2} + 189072q^{4} + O(q^{5}) \]

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)no
Number of roots312
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong A_{12}^{2}\)
Rank of \(R(L)\)24
\([L : R(L)]\)13
\(R(L)\) is primitive in \(L\)no: \(L / R(L) \cong \mathbb{Z}/13\)

Components of \(\Phi(L)\)

ComponentTypeValues of \(b(r, r)\)Number of roots\(\mathbb{Z}\Phi_c\)
\(\Phi_{1}\)\(A_{12}\)\(\{2\}\)156\(\cong A_{12}\)
\(\Phi_{2}\)\(A_{12}\)\(\{2\}\)156\(\cong A_{12}\)

Simple roots

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

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

Notes

The components \(b(e_i, e_j)\) are those of the section GRAM of the entry NIEM_2A12 of the Catalogue of Lattices.

References

  1. G. Nebe and N. J. A. Sloane, Catalogue of Lattices, entry NIEM_2A12. https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/NIEM_2A12.html