Tag 0098
\(U \oplus E_{6}(-1)\)
U + E6(-1)
Invariants
| Rank | 8 |
|---|---|
| Signature \((n_+, n_-)\) | (1, 7) |
| Determinant | \(-3\) |
Gram tensor
The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{8}\).
In this basis, \(L = {}\)\(\mathbb{Z}\{e_{1}, e_{2}\}\) \(\oplus\) \(\mathbb{Z}\{e_{3}, \dots, e_{8}\}\), an orthogonal sum.
| \(e_{1}\) | \(e_{2}\) | \(e_{3}\) | \(e_{4}\) | \(e_{5}\) | \(e_{6}\) | \(e_{7}\) | \(e_{8}\) | |
|---|---|---|---|---|---|---|---|---|
| \(e_{1}\) | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(e_{2}\) | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(e_{3}\) | 0 | 0 | -2 | 0 | 1 | 0 | 0 | 0 |
| \(e_{4}\) | 0 | 0 | 0 | -2 | 0 | 1 | 0 | 0 |
| \(e_{5}\) | 0 | 0 | 1 | 0 | -2 | 1 | 0 | 0 |
| \(e_{6}\) | 0 | 0 | 0 | 1 | 1 | -2 | 1 | 0 |
| \(e_{7}\) | 0 | 0 | 0 | 0 | 0 | 1 | -2 | 1 |
| \(e_{8}\) | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -2 |
[[0, 1, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [0, 0, -2, 0, 1, 0, 0, 0], [0, 0, 0, -2, 0, 1, 0, 0], [0, 0, 1, 0, -2, 1, 0, 0], [0, 0, 0, 1, 1, -2, 1, 0], [0, 0, 0, 0, 0, 1, -2, 1], [0, 0, 0, 0, 0, 0, 1, -2]]
Integral lattice
| Parity | even |
|---|---|
| Discriminant group | \(\mathbb{Z}/3\) |
| Unimodular | no |
| Integral overlattices \(L \subseteq M \subseteq L^*\) | 1 |
| Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\) | \(\{2, 3\}\) |
| Character of the discriminant \(\chi_D\), \(D = (-1)^{4} \det L\) | \(\chi_{-3} = \left(\frac{-3}{\cdot}\right)\) |
| Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\) | \(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 7)\, L^\Sigma(s - 4, \chi_{-3})}{L^\Sigma(s - 3, \chi_{-3})}\) |
| Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\) | \(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 7)}{L^\Sigma(s - 3, \chi_{-3})}\) |
| Genus symbol | \(\mathrm{II}_{1,7}\ \bigl(2\colon 1^{-8};\ 3\colon 1^{7}\,3^{-1}\bigr)\) |
| Class number of the genus | 1 |
| Hyperbolic index \(\max\{n : L \cong U^n \oplus L'\}\) | \(\geq 1\), by an embedding of \(U^{1}\) in the morphism files |
Indefinite lattice
| Isotropic | yes |
|---|
Roots
| Root lattice: \(L = \mathbb{Z}\Phi(L)\) | yes |
|---|---|
| \(L = \mathbb{Z}\Phi_S(L)\) for | \(S = \{-2, 2\}\) |
| \(R(L) = \mathbb{Z}\Phi(L)\) | \(L\) |
| Rank of \(R(L)\) | 8 |
| \([L : R(L)]\) | 1 |
| \(R(L)\) is primitive in \(L\) | yes |
| Proof | Each \(r_i\) in roots that generate \(\mathbb{Z}\Phi(L)\) is a root, and the \(r_i\) generate \(L\). |
Roots that generate \(\mathbb{Z}\Phi(L)\)
The \(r_i\) are a basis of \(L\).
| Root | Coordinates | \(b(r, r)\) |
|---|---|---|
| \(r_{1}\) | \(e_{3}\) | \(-2\) |
| \(r_{2}\) | \(e_{4}\) | \(-2\) |
| \(r_{3}\) | \(e_{5}\) | \(-2\) |
| \(r_{4}\) | \(e_{6}\) | \(-2\) |
| \(r_{5}\) | \(e_{7}\) | \(-2\) |
| \(r_{6}\) | \(e_{8}\) | \(-2\) |
| \(r_{7}\) | \(e_{1} + e_{2}\) | \(2\) |
| \(r_{8}\) | \(e_{1} + e_{3}\) | \(-2\) |