Tag 0153
\(U^{2} \oplus E_{8}(-2)\)
U^2 + E8(-2)
Invariants
| Rank | 12 |
|---|---|
| Signature \((n_+, n_-)\) | (2, 10) |
| Determinant | \(256\) |
Gram tensor
The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{12}\).
In this basis, \(L = {}\)\(\mathbb{Z}\{e_{1}, e_{2}\}\) \(\oplus\) \(\mathbb{Z}\{e_{3}, e_{4}\}\) \(\oplus\) \(\mathbb{Z}\{e_{5}, \dots, e_{12}\}\), an orthogonal sum.
| \(e_{1}\) | \(e_{2}\) | \(e_{3}\) | \(e_{4}\) | \(e_{5}\) | \(e_{6}\) | \(e_{7}\) | \(e_{8}\) | \(e_{9}\) | \(e_{10}\) | \(e_{11}\) | \(e_{12}\) | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(e_{1}\) | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(e_{2}\) | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(e_{3}\) | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(e_{4}\) | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(e_{5}\) | 0 | 0 | 0 | 0 | -4 | 0 | 2 | 0 | 0 | 0 | 0 | 0 |
| \(e_{6}\) | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 2 | 0 | 0 | 0 | 0 |
| \(e_{7}\) | 0 | 0 | 0 | 0 | 2 | 0 | -4 | 2 | 0 | 0 | 0 | 0 |
| \(e_{8}\) | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -4 | 2 | 0 | 0 | 0 |
| \(e_{9}\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -4 | 2 | 0 | 0 |
| \(e_{10}\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -4 | 2 | 0 |
| \(e_{11}\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -4 | 2 |
| \(e_{12}\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -4 |
[[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -4, 0, 2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -4, 0, 2, 0, 0, 0, 0], [0, 0, 0, 0, 2, 0, -4, 2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 2, 2, -4, 2, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 2, -4, 2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, -4, 2, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 2, -4, 2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, -4]]
Integral lattice
| Parity | even |
|---|---|
| Discriminant group | \((\mathbb{Z}/2)^{8}\) |
| Unimodular | no |
| Integral overlattices \(L \subseteq M \subseteq L^*\) | not decided |
| Nikulin invariants \((r, a, \delta)\) | \((12, 8, 0)\) |
| 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 | \(\mathrm{II}_{2,10}\ \bigl(2\colon 1^{4}:2^{8}\bigr)\) |
| Class number of the genus | 1 |
| Hyperbolic index \(\max\{n : L \cong U^n \oplus L'\}\) | \(\geq 2\), by an embedding of \(U^{2}\) in the morphism files |
Indefinite lattice
| Isotropic | yes |
|---|
Roots
| Root lattice: \(L = \mathbb{Z}\Phi(L)\) | yes |
|---|---|
| \(L = \mathbb{Z}\Phi_S(L)\) for | \(S = \{-4, 2\}\) |
| \(R(L) = \mathbb{Z}\Phi(L)\) | \(L\) |
| Rank of \(R(L)\) | 12 |
| \([L : R(L)]\) | 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 | \(b(r, r)\) |
|---|---|---|
| \(r_{1}\) | \(e_{5}\) | \(-4\) |
| \(r_{2}\) | \(e_{6}\) | \(-4\) |
| \(r_{3}\) | \(e_{7}\) | \(-4\) |
| \(r_{4}\) | \(e_{8}\) | \(-4\) |
| \(r_{5}\) | \(e_{9}\) | \(-4\) |
| \(r_{6}\) | \(e_{10}\) | \(-4\) |
| \(r_{7}\) | \(e_{11}\) | \(-4\) |
| \(r_{8}\) | \(e_{12}\) | \(-4\) |
| \(r_{9}\) | \(e_{1} + e_{2}\) | \(2\) |
| \(r_{10}\) | \(e_{3} + e_{4}\) | \(2\) |
| \(r_{11}\) | \(e_{1} + e_{2} + e_{3}\) | \(2\) |
| \(r_{12}\) | \(e_{1} + e_{3} + e_{4}\) | \(2\) |