Tag 0109
\(D_{9}\)
D9
Invariants
| Rank | 9 |
|---|---|
| Signature \((n_+, n_-)\) | (9, 0) |
| Determinant | \(4\) |
Gram tensor
The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{9}\).
| \(e_{1}\) | \(e_{2}\) | \(e_{3}\) | \(e_{4}\) | \(e_{5}\) | \(e_{6}\) | \(e_{7}\) | \(e_{8}\) | \(e_{9}\) | |
|---|---|---|---|---|---|---|---|---|---|
| \(e_{1}\) | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(e_{2}\) | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(e_{3}\) | 0 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 0 |
| \(e_{4}\) | 0 | 0 | -1 | 2 | -1 | 0 | 0 | 0 | 0 |
| \(e_{5}\) | 0 | 0 | 0 | -1 | 2 | -1 | 0 | 0 | 0 |
| \(e_{6}\) | 0 | 0 | 0 | 0 | -1 | 2 | -1 | 0 | 0 |
| \(e_{7}\) | 0 | 0 | 0 | 0 | 0 | -1 | 2 | -1 | -1 |
| \(e_{8}\) | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2 | 0 |
| \(e_{9}\) | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 2 |
[[2, -1, 0, 0, 0, 0, 0, 0, 0], [-1, 2, -1, 0, 0, 0, 0, 0, 0], [0, -1, 2, -1, 0, 0, 0, 0, 0], [0, 0, -1, 2, -1, 0, 0, 0, 0], [0, 0, 0, -1, 2, -1, 0, 0, 0], [0, 0, 0, 0, -1, 2, -1, 0, 0], [0, 0, 0, 0, 0, -1, 2, -1, -1], [0, 0, 0, 0, 0, 0, -1, 2, 0], [0, 0, 0, 0, 0, 0, -1, 0, 2]]
Integral lattice
| Parity | even |
|---|---|
| Discriminant group | \(\mathbb{Z}/4\) |
| Unimodular | no |
| Integral overlattices \(L \subseteq M \subseteq L^*\) | 2 |
| Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\) | \(\{2\}\) |
| Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\) | \(\zeta^\Sigma(X_0, s) = \zeta^\Sigma(s - 8)\) |
| Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\) | \(\zeta^\Sigma(X_n, s) = \zeta^\Sigma(s - 8)\, L^\Sigma(s - 4, \chi_{4n})\) |
| Genus symbol | \(\mathrm{II}_{9,0}\ \bigl(2\colon 1^{8}:[4^{1}]_{1}\bigr)\) |
| Class number of the genus | 2 |
Definite lattice
| Minimum of \(b(x, x)\), \(x \neq 0\) | \(2\) |
|---|---|
| Kissing number | 144 |
| Order of the isometry group | \(185794560 = 2^{16} \cdot 3^{4} \cdot 5 \cdot 7\) |
Theta series
\[ \sum_{x \in L} q^{b(x, x)} = 1 + 144q^{2} + 2034q^{4} + 7392q^{6} + O(q^{7}) \]Roots
| Root lattice: \(L = \mathbb{Z}\Phi(L)\) | yes |
|---|---|
| \(L = \mathbb{Z}\Phi_S(L)\) for | \(S = \{2\}\) |
| Number of roots | 162 |
| \(R(L) = \mathbb{Z}\Phi(L)\) | \(\cong D_{9}\) |
| Rank of \(R(L)\) | 9 |
| \([L : R(L)]\) | 1 |
| \(R(L)\) is primitive in \(L\) | yes |
Components of \(\Phi(L)\)
| Component | Type | Values of \(b(r, r)\) | Number of roots | \(\mathbb{Z}\Phi_c\) |
|---|---|---|---|---|
| \(\Phi_{1}\) | \(C_{9}\) | \(\{2, 4\}\) | 162 | \(\cong D_{9}\) |
Simple roots
| Simple root | Coordinates | \(b(r, r)\) |
|---|---|---|
| \(\alpha_{1}\) | \(e_{1}\) | \(2\) |
| \(\alpha_{2}\) | \(e_{2}\) | \(2\) |
| \(\alpha_{3}\) | \(e_{3}\) | \(2\) |
| \(\alpha_{4}\) | \(e_{4}\) | \(2\) |
| \(\alpha_{5}\) | \(e_{5}\) | \(2\) |
| \(\alpha_{6}\) | \(e_{6}\) | \(2\) |
| \(\alpha_{7}\) | \(e_{7}\) | \(2\) |
| \(\alpha_{8}\) | \(e_{9}\) | \(2\) |
| \(\alpha_{9}\) | \(e_{8} - e_{9}\) | \(4\) |