Tag 0107
\(\mathrm{I}_{9,0}\)
I_{9,0}
Invariants
| Rank | 9 |
|---|---|
| Signature \((n_+, n_-)\) | (9, 0) |
| Determinant | \(1\) |
Gram tensor
The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{9}\).
In this basis, \(L = {}\)\(\mathbb{Z}\{e_{1}\}\) \(\oplus\) \(\mathbb{Z}\{e_{2}\}\) \(\oplus\) \(\mathbb{Z}\{e_{3}\}\) \(\oplus\) \(\mathbb{Z}\{e_{4}\}\) \(\oplus\) \(\mathbb{Z}\{e_{5}\}\) \(\oplus\) \(\mathbb{Z}\{e_{6}\}\) \(\oplus\) \(\mathbb{Z}\{e_{7}\}\) \(\oplus\) \(\mathbb{Z}\{e_{8}\}\) \(\oplus\) \(\mathbb{Z}\{e_{9}\}\), an orthogonal sum.
| \(e_{1}\) | \(e_{2}\) | \(e_{3}\) | \(e_{4}\) | \(e_{5}\) | \(e_{6}\) | \(e_{7}\) | \(e_{8}\) | \(e_{9}\) | |
|---|---|---|---|---|---|---|---|---|---|
| \(e_{1}\) | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(e_{2}\) | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(e_{3}\) | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(e_{4}\) | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| \(e_{5}\) | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| \(e_{6}\) | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| \(e_{7}\) | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| \(e_{8}\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| \(e_{9}\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
[[1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1]]
Integral lattice
| Parity | odd |
|---|---|
| Discriminant group | \(0\) |
| Unimodular | yes |
| Integral overlattices \(L \subseteq M \subseteq L^*\) | 1 |
| Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\) | \(\{2\}\) |
| 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 \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\) | \(\zeta^\Sigma(X_n, s) = \zeta^\Sigma(s - 8)\, L^\Sigma(s - 4, \chi_{n})\) |
| Genus symbol | \(\mathrm{I}_{9,0}\) |
| Class number of the genus | 2 |
Definite lattice
| Minimum of \(b(x, x)\), \(x \neq 0\) | \(1\) |
|---|---|
| Kissing number | 18 |
| Order of the isometry group | \(185794560 = 2^{16} \cdot 3^{4} \cdot 5 \cdot 7\) |
Theta series
\[ \sum_{x \in L} q^{b(x, x)} = 1 + 18q + 144q^{2} + 672q^{3} + 2034q^{4} + 4320q^{5} + 7392q^{6} + O(q^{7}) \]Roots
| Root lattice: \(L = \mathbb{Z}\Phi(L)\) | yes |
|---|---|
| \(L = \mathbb{Z}\Phi_S(L)\) for | \(S = \{1\}\) |
| Number of roots | 162 |
| \(R(L) = \mathbb{Z}\Phi(L)\) | \(\cong \mathrm{I}_{9,0}\) |
| 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 | \(\mathbb{Z}\Phi_c\) |
|---|---|---|---|---|
| \(\Phi_{1}\) | \(B_{9}\) | \(\{1, 2\}\) | 162 | \(\cong \mathrm{I}_{9,0}\) |
Simple roots
| Simple root | Coordinates | \(b(r, r)\) |
|---|---|---|
| \(\alpha_{1}\) | \(e_{1} - e_{2}\) | \(2\) |
| \(\alpha_{2}\) | \(e_{2} - e_{3}\) | \(2\) |
| \(\alpha_{3}\) | \(e_{3} - e_{4}\) | \(2\) |
| \(\alpha_{4}\) | \(e_{4} - e_{5}\) | \(2\) |
| \(\alpha_{5}\) | \(e_{5} - e_{6}\) | \(2\) |
| \(\alpha_{6}\) | \(e_{6} - e_{7}\) | \(2\) |
| \(\alpha_{7}\) | \(e_{7} - e_{8}\) | \(2\) |
| \(\alpha_{8}\) | \(e_{8} - e_{9}\) | \(2\) |
| \(\alpha_{9}\) | \(e_{9}\) | \(1\) |