Tag 0298
\(N(D_{12}^{2})\)
N(D12^2)·NIEM_2D12·Niemeier lattice with root system N(D12^2)
Invariants
| Rank | 24 |
|---|---|
| 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 \(e_{1}, \dots, e_{24}\).
| \(e_{1}\) | \(e_{2}\) | \(e_{3}\) | \(e_{4}\) | \(e_{5}\) | \(e_{6}\) | \(e_{7}\) | \(e_{8}\) | \(e_{9}\) | \(e_{10}\) | \(e_{11}\) | \(e_{12}\) | \(e_{13}\) | \(e_{14}\) | \(e_{15}\) | \(e_{16}\) | \(e_{17}\) | \(e_{18}\) | \(e_{19}\) | \(e_{20}\) | \(e_{21}\) | \(e_{22}\) | \(e_{23}\) | \(e_{24}\) | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(e_{1}\) | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |
| \(e_{2}\) | -1 | 2 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 |
| \(e_{3}\) | 0 | 1 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 |
| \(e_{4}\) | 0 | -1 | -1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 1 |
| \(e_{5}\) | 0 | -1 | -1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 1 |
| \(e_{6}\) | 0 | -1 | -1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 1 |
| \(e_{7}\) | 0 | -1 | -1 | 1 | 1 | 1 | 2 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
| \(e_{8}\) | 0 | -1 | -1 | 1 | 1 | 1 | 0 | 2 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
| \(e_{9}\) | 0 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 1 |
| \(e_{10}\) | 0 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
| \(e_{11}\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 |
| \(e_{12}\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 2 | 1 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | 0 |
| \(e_{13}\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 0 | -1 |
| \(e_{14}\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 1 |
| \(e_{15}\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 0 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | -1 | 0 |
| \(e_{16}\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 1 |
| \(e_{17}\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 0 | 0 | -1 | 1 |
| \(e_{18}\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 0 | 0 | 0 | 0 |
| \(e_{19}\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 0 | 0 | 0 | 1 |
| \(e_{20}\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 0 | 0 | 0 | 0 |
| \(e_{21}\) | 1 | -1 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | -1 | 0 | 1 |
| \(e_{22}\) | 1 | -1 | -1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 4 | 1 | 0 |
| \(e_{23}\) | 0 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | -1 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | 4 | 1 |
| \(e_{24}\) | 0 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 0 | -1 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 4 |
[[2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0], [-1, 2, 1, -1, -1, -1, -1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, -1, -1], [0, 1, 2, -1, -1, -1, -1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, -1], [0, -1, -1, 2, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1], [0, -1, -1, 1, 2, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1], [0, -1, -1, 1, 1, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1], [0, -1, -1, 1, 1, 1, 2, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1], [0, -1, -1, 1, 1, 1, 0, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, -1, -1, 1, 1, 1, 1, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1], [0, -1, -1, 1, 1, 1, 1, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1, 1, 1, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, -1, 0, 0, 0, 0, 0, -1, 1, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 2, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0, 1, 2, 1, 1, 1, 1, 1, 0, 0, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, 1, 2, 1, 1, 1, 1, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, 1, 1, 2, 1, 1, 1, 0, 0, -1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, 1, 1, 1, 2, 1, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, 1, 1, 1, 1, 2, 1, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, 1, 1, 1, 1, 1, 2, 0, 0, 0, 0], [1, -1, 0, 1, 1, 1, 1, 0, 1, 1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 0, 4, -1, 0, 1], [1, -1, -1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, -1, 4, 1, 0], [0, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, -1, 0, -1, 0, 0, 0, 0, 1, 4, 1], [0, -1, -1, 1, 1, 1, 1, 1, 1, 1, -1, 0, -1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 4]]
Integral lattice
| Parity | even |
|---|---|
| Discriminant group | \(0\) |
| Unimodular | yes |
| 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 number | 528 |
Theta series
\[ \sum_{x \in L} q^{b(x, x)} = 1 + 528q^{2} + 183888q^{4} + O(q^{5}) \]Roots
| Root lattice: \(L = \mathbb{Z}\Phi(L)\) | no |
|---|---|
| Number of roots | 528 |
| \(R(L) = \mathbb{Z}\Phi(L)\) | \(\cong D_{12}^{2}\) |
| Rank of \(R(L)\) | 24 |
| \([L : R(L)]\) | 4 |
| \(R(L)\) is primitive in \(L\) | no: \(L / R(L) \cong (\mathbb{Z}/2)^{2}\) |
Components of \(\Phi(L)\)
| Component | Type | Values of \(b(r, r)\) | Number of roots | \(\mathbb{Z}\Phi_c\) |
|---|---|---|---|---|
| \(\Phi_{1}\) | \(D_{12}\) | \(\{2\}\) | 264 | \(\cong D_{12}\) |
| \(\Phi_{2}\) | \(D_{12}\) | \(\{2\}\) | 264 | \(\cong D_{12}\) |
Simple roots
\(\alpha^{(c)}_j\) is simple root \(j\) of the component \(\Phi_c\).
| Simple root | Coordinates | \(b(r, r)\) |
|---|---|---|
| \(\alpha^{(1)}_{1}\) | \(e_{7} + e_{8} - e_{13} + e_{14} - e_{15} + e_{16} + e_{17} - e_{18} + e_{19} - e_{20} - 2e_{24}\) | \(2\) |
| \(\alpha^{(1)}_{2}\) | \(e_{11} - e_{12} - e_{15} - e_{17} + e_{18} + e_{20} - e_{23} + e_{24}\) | \(2\) |
| \(\alpha^{(1)}_{3}\) | \(e_{13}\) | \(2\) |
| \(\alpha^{(1)}_{4}\) | \(e_{12} - e_{13} - e_{14} + e_{15}\) | \(2\) |
| \(\alpha^{(1)}_{5}\) | \(e_{14} - e_{15}\) | \(2\) |
| \(\alpha^{(1)}_{6}\) | \(e_{15} - e_{16}\) | \(2\) |
| \(\alpha^{(1)}_{7}\) | \(e_{16} - e_{17}\) | \(2\) |
| \(\alpha^{(1)}_{8}\) | \(e_{17} - e_{18}\) | \(2\) |
| \(\alpha^{(1)}_{9}\) | \(e_{18} - e_{19}\) | \(2\) |
| \(\alpha^{(1)}_{10}\) | \(e_{19} - e_{20}\) | \(2\) |
| \(\alpha^{(1)}_{11}\) | \(e_{11} - e_{12} - e_{15} + e_{20}\) | \(2\) |
| \(\alpha^{(1)}_{12}\) | \(e_{20}\) | \(2\) |
| \(\alpha^{(2)}_{1}\) | \(e_{2} + e_{8} - e_{10} + e_{21} + e_{22}\) | \(2\) |
| \(\alpha^{(2)}_{2}\) | \(e_{1} - e_{2} - e_{8} + e_{10} - e_{21} - e_{22}\) | \(2\) |
| \(\alpha^{(2)}_{3}\) | \(e_{2} - e_{3} - e_{4} - e_{5} - e_{6} + 2e_{7} + 2e_{8} - e_{9} + e_{11} + e_{12} + e_{15} + e_{21} - e_{22}\) | \(2\) |
| \(\alpha^{(2)}_{4}\) | \(e_{4} + e_{5} + e_{6} - 2e_{7} - 2e_{8} + e_{9} - e_{11} - e_{12} - e_{15} - e_{21} + e_{22}\) | \(2\) |
| \(\alpha^{(2)}_{5}\) | \(e_{3} - e_{4} + e_{7} + e_{8}\) | \(2\) |
| \(\alpha^{(2)}_{6}\) | \(e_{4} - e_{5}\) | \(2\) |
| \(\alpha^{(2)}_{7}\) | \(e_{5} - e_{6}\) | \(2\) |
| \(\alpha^{(2)}_{8}\) | \(e_{6} - e_{7} - e_{8}\) | \(2\) |
| \(\alpha^{(2)}_{9}\) | \(e_{10}\) | \(2\) |
| \(\alpha^{(2)}_{10}\) | \(e_{9} - e_{10}\) | \(2\) |
| \(\alpha^{(2)}_{11}\) | \(e_{7} - e_{9}\) | \(2\) |
| \(\alpha^{(2)}_{12}\) | \(e_{8} - e_{9}\) | \(2\) |
Notes
The components \(b(e_i, e_j)\) are
those of the section GRAM of the entry
NIEM_2D12 of the Catalogue of Lattices.
References
- G. Nebe and N. J. A. Sloane, Catalogue of Lattices, entry NIEM_2D12. https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/NIEM_2D12.html