Tag 028T

\(N(A_{1}^{24})\)

N(A1^24)·NIEM_24A1·Niemeier lattice with root system N(A1^24)

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

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)\(e_{13}\)\(e_{14}\)\(e_{15}\)\(e_{16}\)\(e_{17}\)\(e_{18}\)\(e_{19}\)\(e_{20}\)\(e_{21}\)\(e_{22}\)\(e_{23}\)\(e_{24}\)
\(e_{1}\)200000000111111111000000
\(e_{2}\)02000000010011-1-100100110
\(e_{3}\)0020000001-1-1-10-10-10000-1-11
\(e_{4}\)0002000000-11001101-1-100-1-1
\(e_{5}\)000020000001110000010111
\(e_{6}\)0000020000-1011-11101110-11
\(e_{7}\)000000200-1000-1101-1-101-10-1
\(e_{8}\)0000000200100100-1-1-10-1-1-10
\(e_{9}\)000000002-101-100111-1-1-110-1
\(e_{10}\)111000-10-141012-1-1-1020-1002
\(e_{11}\)10-1-10-1010141111-10-10-1-2010
\(e_{12}\)10-1110001014111222-1-1-120-1
\(e_{13}\)11-101100-1111420010211111
\(e_{14}\)110011-11021124-100011-1102
\(e_{15}\)1-1-110-1100-1110-14110-2-10-10-2
\(e_{16}\)1-10101001-1-12001422-1010-2-1
\(e_{17}\)10-10011-11-10210124100110-1
\(e_{18}\)100100-1-110-120002140-1020-1
\(e_{19}\)010-101-1-1-120-121-2-100411112
\(e_{20}\)000-11100-10-1-111-100-1142012
\(e_{21}\)0000011-1-1-1-2-11-10110124-100
\(e_{22}\)01-1010-1-1100211-101210-1420
\(e_{23}\)01-1-11-10-10010100-200110240
\(e_{24}\)001-111-10-120-112-2-1-1-1220004
 
[[2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, -1, -1, 0, 0, 1, 0, 0, 1, 1, 0], [0, 0, 2, 0, 0, 0, 0, 0, 0, 1, -1, -1, -1, 0, -1, 0, -1, 0, 0, 0, 0, -1, -1, 1], [0, 0, 0, 2, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 1, 1, 0, 1, -1, -1, 0, 0, -1, -1], [0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1], [0, 0, 0, 0, 0, 2, 0, 0, 0, 0, -1, 0, 1, 1, -1, 1, 1, 0, 1, 1, 1, 0, -1, 1], [0, 0, 0, 0, 0, 0, 2, 0, 0, -1, 0, 0, 0, -1, 1, 0, 1, -1, -1, 0, 1, -1, 0, -1], [0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, -1, -1, -1, 0, -1, -1, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, -1, 0, 1, -1, 0, 0, 1, 1, 1, -1, -1, -1, 1, 0, -1], [1, 1, 1, 0, 0, 0, -1, 0, -1, 4, 1, 0, 1, 2, -1, -1, -1, 0, 2, 0, -1, 0, 0, 2], [1, 0, -1, -1, 0, -1, 0, 1, 0, 1, 4, 1, 1, 1, 1, -1, 0, -1, 0, -1, -2, 0, 1, 0], [1, 0, -1, 1, 1, 0, 0, 0, 1, 0, 1, 4, 1, 1, 1, 2, 2, 2, -1, -1, -1, 2, 0, -1], [1, 1, -1, 0, 1, 1, 0, 0, -1, 1, 1, 1, 4, 2, 0, 0, 1, 0, 2, 1, 1, 1, 1, 1], [1, 1, 0, 0, 1, 1, -1, 1, 0, 2, 1, 1, 2, 4, -1, 0, 0, 0, 1, 1, -1, 1, 0, 2], [1, -1, -1, 1, 0, -1, 1, 0, 0, -1, 1, 1, 0, -1, 4, 1, 1, 0, -2, -1, 0, -1, 0, -2], [1, -1, 0, 1, 0, 1, 0, 0, 1, -1, -1, 2, 0, 0, 1, 4, 2, 2, -1, 0, 1, 0, -2, -1], [1, 0, -1, 0, 0, 1, 1, -1, 1, -1, 0, 2, 1, 0, 1, 2, 4, 1, 0, 0, 1, 1, 0, -1], [1, 0, 0, 1, 0, 0, -1, -1, 1, 0, -1, 2, 0, 0, 0, 2, 1, 4, 0, -1, 0, 2, 0, -1], [0, 1, 0, -1, 0, 1, -1, -1, -1, 2, 0, -1, 2, 1, -2, -1, 0, 0, 4, 1, 1, 1, 1, 2], [0, 0, 0, -1, 1, 1, 0, 0, -1, 0, -1, -1, 1, 1, -1, 0, 0, -1, 1, 4, 2, 0, 1, 2], [0, 0, 0, 0, 0, 1, 1, -1, -1, -1, -2, -1, 1, -1, 0, 1, 1, 0, 1, 2, 4, -1, 0, 0], [0, 1, -1, 0, 1, 0, -1, -1, 1, 0, 0, 2, 1, 1, -1, 0, 1, 2, 1, 0, -1, 4, 2, 0], [0, 1, -1, -1, 1, -1, 0, -1, 0, 0, 1, 0, 1, 0, 0, -2, 0, 0, 1, 1, 0, 2, 4, 0], [0, 0, 1, -1, 1, 1, -1, 0, -1, 2, 0, -1, 1, 2, -2, -1, -1, -1, 2, 2, 0, 0, 0, 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 number48

Theta series

\[ \sum_{x \in L} q^{b(x, x)} = 1 + 48q^{2} + 195408q^{4} + O(q^{5}) \]

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)no
Number of roots48
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong \langle 2 \rangle^{24}\)
Rank of \(R(L)\)24
\([L : R(L)]\)4096
\(R(L)\) is primitive in \(L\)no: \(L / R(L) \cong (\mathbb{Z}/2)^{12}\)

Components of \(\Phi(L)\)

ComponentTypeValues of \(b(r, r)\)Number of roots\(\mathbb{Z}\Phi_c\)
\(\Phi_{1}\)\(A_{1}\)\(\{2\}\)2\(\cong \langle 2 \rangle\)
\(\Phi_{2}\)\(A_{1}\)\(\{2\}\)2\(\cong \langle 2 \rangle\)
\(\Phi_{3}\)\(A_{1}\)\(\{2\}\)2\(\cong \langle 2 \rangle\)
\(\Phi_{4}\)\(A_{1}\)\(\{2\}\)2\(\cong \langle 2 \rangle\)
\(\Phi_{5}\)\(A_{1}\)\(\{2\}\)2\(\cong \langle 2 \rangle\)
\(\Phi_{6}\)\(A_{1}\)\(\{2\}\)2\(\cong \langle 2 \rangle\)
\(\Phi_{7}\)\(A_{1}\)\(\{2\}\)2\(\cong \langle 2 \rangle\)
\(\Phi_{8}\)\(A_{1}\)\(\{2\}\)2\(\cong \langle 2 \rangle\)
\(\Phi_{9}\)\(A_{1}\)\(\{2\}\)2\(\cong \langle 2 \rangle\)
\(\Phi_{10}\)\(A_{1}\)\(\{2\}\)2\(\cong \langle 2 \rangle\)
\(\Phi_{11}\)\(A_{1}\)\(\{2\}\)2\(\cong \langle 2 \rangle\)
\(\Phi_{12}\)\(A_{1}\)\(\{2\}\)2\(\cong \langle 2 \rangle\)
\(\Phi_{13}\)\(A_{1}\)\(\{2\}\)2\(\cong \langle 2 \rangle\)
\(\Phi_{14}\)\(A_{1}\)\(\{2\}\)2\(\cong \langle 2 \rangle\)
\(\Phi_{15}\)\(A_{1}\)\(\{2\}\)2\(\cong \langle 2 \rangle\)
\(\Phi_{16}\)\(A_{1}\)\(\{2\}\)2\(\cong \langle 2 \rangle\)
\(\Phi_{17}\)\(A_{1}\)\(\{2\}\)2\(\cong \langle 2 \rangle\)
\(\Phi_{18}\)\(A_{1}\)\(\{2\}\)2\(\cong \langle 2 \rangle\)
\(\Phi_{19}\)\(A_{1}\)\(\{2\}\)2\(\cong \langle 2 \rangle\)
\(\Phi_{20}\)\(A_{1}\)\(\{2\}\)2\(\cong \langle 2 \rangle\)
\(\Phi_{21}\)\(A_{1}\)\(\{2\}\)2\(\cong \langle 2 \rangle\)
\(\Phi_{22}\)\(A_{1}\)\(\{2\}\)2\(\cong \langle 2 \rangle\)
\(\Phi_{23}\)\(A_{1}\)\(\{2\}\)2\(\cong \langle 2 \rangle\)
\(\Phi_{24}\)\(A_{1}\)\(\{2\}\)2\(\cong \langle 2 \rangle\)

Simple roots

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

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

Notes

The components \(b(e_i, e_j)\) are those of the section GRAM of the entry NIEM_24A1 of the Catalogue of Lattices.

References

  1. G. Nebe and N. J. A. Sloane, Catalogue of Lattices, entry NIEM_24A1. https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/NIEM_24A1.html