Tag 0151
\(D_{12}^{+}\)
D12+
Invariants
| Rank | 12 |
|---|---|
| Signature \((n_+, n_-)\) | (12, 0) |
| Determinant | \(1\) |
Gram tensor
The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{12}\).
| \(e_{1}\) | \(e_{2}\) | \(e_{3}\) | \(e_{4}\) | \(e_{5}\) | \(e_{6}\) | \(e_{7}\) | \(e_{8}\) | \(e_{9}\) | \(e_{10}\) | \(e_{11}\) | \(e_{12}\) | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(e_{1}\) | 2 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 |
| \(e_{2}\) | 1 | 2 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 |
| \(e_{3}\) | 1 | 1 | 2 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 |
| \(e_{4}\) | -1 | 0 | -1 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 |
| \(e_{5}\) | 1 | 1 | 1 | 0 | 2 | 1 | 1 | 1 | 1 | 1 | 0 | -1 |
| \(e_{6}\) | 1 | 1 | 1 | 0 | 1 | 2 | 1 | 1 | 1 | 1 | 0 | -1 |
| \(e_{7}\) | 1 | 1 | 1 | 0 | 1 | 1 | 2 | 1 | 1 | 1 | 0 | -1 |
| \(e_{8}\) | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 2 | 1 | 1 | 0 | -1 |
| \(e_{9}\) | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 2 | 1 | 0 | -1 |
| \(e_{10}\) | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 3 | 1 | -1 |
| \(e_{11}\) | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 3 | 0 |
| \(e_{12}\) | -1 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 3 |
[[2, 1, 1, -1, 1, 1, 1, 1, 1, 1, -1, -1], [1, 2, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0], [1, 1, 2, -1, 1, 1, 1, 1, 1, 1, 0, 0], [-1, 0, -1, 2, 0, 0, 0, 0, 0, -1, 0, 0], [1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 0, -1], [1, 1, 1, 0, 1, 2, 1, 1, 1, 1, 0, -1], [1, 1, 1, 0, 1, 1, 2, 1, 1, 1, 0, -1], [1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 0, -1], [1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 0, -1], [1, 1, 1, -1, 1, 1, 1, 1, 1, 3, 1, -1], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 0], [-1, 0, 0, 0, -1, -1, -1, -1, -1, -1, 0, 3]]
Integral lattice
| Parity | odd |
|---|---|
| Discriminant group | \(0\) |
| Unimodular | yes |
| Integral overlattices \(L \subseteq M \subseteq L^*\) | 1 |
| Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\) | \(\{2\}\) |
| Character of the discriminant \(\chi_D\), \(D = (-1)^{6} \det L\) | trivial |
| Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\) | \(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 11)\, \zeta^\Sigma(s - 6)}{\zeta^\Sigma(s - 5)}\) |
| Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\) | \(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 11)}{\zeta^\Sigma(s - 5)}\) |
| Genus symbol | \(\mathrm{I}_{12,0}\) |
| Class number of the genus | 3 |
Definite lattice
| Minimum of \(b(x, x)\), \(x \neq 0\) | \(2\) |
|---|---|
| Kissing number | 264 |
| Order of the isometry group | \(980995276800 = 2^{21} \cdot 3^{5} \cdot 5^{2} \cdot 7 \cdot 11\) |
Theta series
\[ \sum_{x \in L} q^{b(x, x)} = 1 + 264q^{2} + 2048q^{3} + 7944q^{4} + 24576q^{5} + 64416q^{6} + O(q^{7}) \]Roots
| Root lattice: \(L = \mathbb{Z}\Phi(L)\) | no |
|---|---|
| Number of roots | 264 |
| \(R(L) = \mathbb{Z}\Phi(L)\) | \(\cong D_{12}\) |
| Rank of \(R(L)\) | 12 |
| \([L : R(L)]\) | 2 |
| \(R(L)\) is primitive in \(L\) | no: \(L / R(L) \cong \mathbb{Z}/2\) |
Components of \(\Phi(L)\)
| Component | Type | Values of \(b(r, r)\) | Number of roots | \(\mathbb{Z}\Phi_c\) |
|---|---|---|---|---|
| \(\Phi_{1}\) | \(D_{12}\) | \(\{2\}\) | 264 | \(\cong D_{12}\) |
Simple roots
| Simple root | Coordinates | \(b(r, r)\) |
|---|---|---|
| \(\alpha_{1}\) | \(e_{1} - e_{2} + e_{4} + e_{10} + e_{12}\) | \(2\) |
| \(\alpha_{2}\) | \(e_{1} - e_{10} + e_{11}\) | \(2\) |
| \(\alpha_{3}\) | \(e_{1} + 2e_{3} + 2e_{4} - e_{5} - e_{6} - e_{7} - e_{8} - e_{9} + e_{10} - e_{12}\) | \(2\) |
| \(\alpha_{4}\) | \(e_{9}\) | \(2\) |
| \(\alpha_{5}\) | \(e_{8} - e_{9}\) | \(2\) |
| \(\alpha_{6}\) | \(e_{7} - e_{8}\) | \(2\) |
| \(\alpha_{7}\) | \(e_{6} - e_{7}\) | \(2\) |
| \(\alpha_{8}\) | \(e_{5} - e_{6}\) | \(2\) |
| \(\alpha_{9}\) | \(e_{3} - e_{5}\) | \(2\) |
| \(\alpha_{10}\) | \(e_{4}\) | \(2\) |
| \(\alpha_{11}\) | \(e_{1} - e_{2}\) | \(2\) |
| \(\alpha_{12}\) | \(e_{2} - e_{3} - e_{4}\) | \(2\) |
Notes
The basis is an LLL-reduced basis.