Tag 028U

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

N(A2^12)·NIEM_12A2·Niemeier lattice with root system N(A2^12)

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

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)\(e_{13}\)\(e_{14}\)\(e_{15}\)\(e_{16}\)\(e_{17}\)\(e_{18}\)\(e_{19}\)\(e_{20}\)\(e_{21}\)\(e_{22}\)\(e_{23}\)\(e_{24}\)
\(e_{1}\)200000000111111110000000
\(e_{2}\)0200000001111-10001111111
\(e_{3}\)0020000001000-10-1-111-1-1010
\(e_{4}\)00020000001-11-11-101-110-1-10
\(e_{5}\)00002000001000-1-110-1-1-1-101
\(e_{6}\)000002100-10011-111-1-1-11-1-1-1
\(e_{7}\)000001200-10011-1000-1-11-1-10
\(e_{8}\)00000002-11-1-1101-1-1-10011-11
\(e_{9}\)0000000-12-101-10011100-1-10-1
\(e_{10}\)11100-1-11-14111-12-1-11210211
\(e_{11}\)1101100-10141100021010-110
\(e_{12}\)110-1000-1111400021110011-1
\(e_{13}\)11010111-1110401000-1020-11
\(e_{14}\)1-1-1-101100-10004-122-2-1-11-1-1-1
\(e_{15}\)1001-1-1-1102001-140-111201-10
\(e_{16}\)10-1-1-110-11-10202042-100100-2
\(e_{17}\)10-10110-11-12102-124-1-100-20-1
\(e_{18}\)01110-10-111110-21-1-1411-1011
\(e_{19}\)011-1-1-1-100201-1-110-11410220
\(e_{20}\)01-11-1-1-1001100-12001141100
\(e_{21}\)01-10-1111-100021010-10141-10
\(e_{22}\)010-1-1-1-11-12-110-110-20211411
\(e_{23}\)011-10-1-1-10111-1-1-100120-1141
\(e_{24}\)01001-101-110-11-10-2-11000114
 
[[2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, -1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1], [0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0, -1, -1, 1, 1, -1, -1, 0, 1, 0], [0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, -1, 1, -1, 1, -1, 0, 1, -1, 1, 0, -1, -1, 0], [0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1, -1, 1, 0, -1, -1, -1, -1, 0, 1], [0, 0, 0, 0, 0, 2, 1, 0, 0, -1, 0, 0, 1, 1, -1, 1, 1, -1, -1, -1, 1, -1, -1, -1], [0, 0, 0, 0, 0, 1, 2, 0, 0, -1, 0, 0, 1, 1, -1, 0, 0, 0, -1, -1, 1, -1, -1, 0], [0, 0, 0, 0, 0, 0, 0, 2, -1, 1, -1, -1, 1, 0, 1, -1, -1, -1, 0, 0, 1, 1, -1, 1], [0, 0, 0, 0, 0, 0, 0, -1, 2, -1, 0, 1, -1, 0, 0, 1, 1, 1, 0, 0, -1, -1, 0, -1], [1, 1, 1, 0, 0, -1, -1, 1, -1, 4, 1, 1, 1, -1, 2, -1, -1, 1, 2, 1, 0, 2, 1, 1], [1, 1, 0, 1, 1, 0, 0, -1, 0, 1, 4, 1, 1, 0, 0, 0, 2, 1, 0, 1, 0, -1, 1, 0], [1, 1, 0, -1, 0, 0, 0, -1, 1, 1, 1, 4, 0, 0, 0, 2, 1, 1, 1, 0, 0, 1, 1, -1], [1, 1, 0, 1, 0, 1, 1, 1, -1, 1, 1, 0, 4, 0, 1, 0, 0, 0, -1, 0, 2, 0, -1, 1], [1, -1, -1, -1, 0, 1, 1, 0, 0, -1, 0, 0, 0, 4, -1, 2, 2, -2, -1, -1, 1, -1, -1, -1], [1, 0, 0, 1, -1, -1, -1, 1, 0, 2, 0, 0, 1, -1, 4, 0, -1, 1, 1, 2, 0, 1, -1, 0], [1, 0, -1, -1, -1, 1, 0, -1, 1, -1, 0, 2, 0, 2, 0, 4, 2, -1, 0, 0, 1, 0, 0, -2], [1, 0, -1, 0, 1, 1, 0, -1, 1, -1, 2, 1, 0, 2, -1, 2, 4, -1, -1, 0, 0, -2, 0, -1], [0, 1, 1, 1, 0, -1, 0, -1, 1, 1, 1, 1, 0, -2, 1, -1, -1, 4, 1, 1, -1, 0, 1, 1], [0, 1, 1, -1, -1, -1, -1, 0, 0, 2, 0, 1, -1, -1, 1, 0, -1, 1, 4, 1, 0, 2, 2, 0], [0, 1, -1, 1, -1, -1, -1, 0, 0, 1, 1, 0, 0, -1, 2, 0, 0, 1, 1, 4, 1, 1, 0, 0], [0, 1, -1, 0, -1, 1, 1, 1, -1, 0, 0, 0, 2, 1, 0, 1, 0, -1, 0, 1, 4, 1, -1, 0], [0, 1, 0, -1, -1, -1, -1, 1, -1, 2, -1, 1, 0, -1, 1, 0, -2, 0, 2, 1, 1, 4, 1, 1], [0, 1, 1, -1, 0, -1, -1, -1, 0, 1, 1, 1, -1, -1, -1, 0, 0, 1, 2, 0, -1, 1, 4, 1], [0, 1, 0, 0, 1, -1, 0, 1, -1, 1, 0, -1, 1, -1, 0, -2, -1, 1, 0, 0, 0, 1, 1, 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 number72

Theta series

\[ \sum_{x \in L} q^{b(x, x)} = 1 + 72q^{2} + 194832q^{4} + O(q^{5}) \]

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)no
Number of roots72
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong A_{2}^{12}\)
Rank of \(R(L)\)24
\([L : R(L)]\)729
\(R(L)\) is primitive in \(L\)no: \(L / R(L) \cong (\mathbb{Z}/3)^{6}\)

Components of \(\Phi(L)\)

ComponentTypeValues of \(b(r, r)\)Number of roots\(\mathbb{Z}\Phi_c\)
\(\Phi_{1}\)\(A_{2}\)\(\{2\}\)6\(\cong A_{2}\)
\(\Phi_{2}\)\(A_{2}\)\(\{2\}\)6\(\cong A_{2}\)
\(\Phi_{3}\)\(A_{2}\)\(\{2\}\)6\(\cong A_{2}\)
\(\Phi_{4}\)\(A_{2}\)\(\{2\}\)6\(\cong A_{2}\)
\(\Phi_{5}\)\(A_{2}\)\(\{2\}\)6\(\cong A_{2}\)
\(\Phi_{6}\)\(A_{2}\)\(\{2\}\)6\(\cong A_{2}\)
\(\Phi_{7}\)\(A_{2}\)\(\{2\}\)6\(\cong A_{2}\)
\(\Phi_{8}\)\(A_{2}\)\(\{2\}\)6\(\cong A_{2}\)
\(\Phi_{9}\)\(A_{2}\)\(\{2\}\)6\(\cong A_{2}\)
\(\Phi_{10}\)\(A_{2}\)\(\{2\}\)6\(\cong A_{2}\)
\(\Phi_{11}\)\(A_{2}\)\(\{2\}\)6\(\cong A_{2}\)
\(\Phi_{12}\)\(A_{2}\)\(\{2\}\)6\(\cong A_{2}\)

Simple roots

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

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

Notes

The components \(b(e_i, e_j)\) are those of the section GRAM of the entry NIEM_12A2 of the Catalogue of Lattices.

References

  1. G. Nebe and N. J. A. Sloane, Catalogue of Lattices, entry NIEM_12A2. https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/NIEM_12A2.html