Tag 0159
\(E_{8} \oplus D_{4}\)
E8 + D4
Invariants
| Rank | 12 |
|---|---|
| Signature \((n_+, n_-)\) | (12, 0) |
| Determinant | \(4\) |
Gram tensor
The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{12}\).
In this basis, \(L = {}\)\(\mathbb{Z}\{e_{1}, \dots, e_{8}\}\) \(\oplus\) \(\mathbb{Z}\{e_{9}, \dots, e_{12}\}\), an orthogonal sum.
| \(e_{1}\) | \(e_{2}\) | \(e_{3}\) | \(e_{4}\) | \(e_{5}\) | \(e_{6}\) | \(e_{7}\) | \(e_{8}\) | \(e_{9}\) | \(e_{10}\) | \(e_{11}\) | \(e_{12}\) | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(e_{1}\) | 2 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(e_{2}\) | 0 | 2 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(e_{3}\) | -1 | 0 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(e_{4}\) | 0 | -1 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(e_{5}\) | 0 | 0 | 0 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(e_{6}\) | 0 | 0 | 0 | 0 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 0 |
| \(e_{7}\) | 0 | 0 | 0 | 0 | 0 | -1 | 2 | -1 | 0 | 0 | 0 | 0 |
| \(e_{8}\) | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2 | 0 | 0 | 0 | 0 |
| \(e_{9}\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -1 | 0 | 0 |
| \(e_{10}\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2 | -1 | -1 |
| \(e_{11}\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2 | 0 |
| \(e_{12}\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 2 |
[[2, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, -1, 2, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 2, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 2, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, 2, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 2, -1, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 2, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 2]]
Integral lattice
| Parity | even |
|---|---|
| Discriminant group | \((\mathbb{Z}/2)^{2}\) |
| Unimodular | no |
| Integral overlattices \(L \subseteq M \subseteq L^*\) | 4 |
| Nikulin invariants \((r, a, \delta)\) | \((12, 2, 0)\) |
| 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 | \(\mathrm{II}_{12,0}\ \bigl(2\colon 1^{-10}:2^{-2}\bigr)\) |
| Class number of the genus | 2 |
Definite lattice
| Minimum of \(b(x, x)\), \(x \neq 0\) | \(2\) |
|---|---|
| Kissing number | 264 |
| Order of the isometry group | \(802632499200 = 2^{21} \cdot 3^{7} \cdot 5^{2} \cdot 7\) |
Theta series
\[ \sum_{x \in L} q^{b(x, x)} = 1 + 264q^{2} + 7944q^{4} + 64416q^{6} + O(q^{7}) \]Roots
| Root lattice: \(L = \mathbb{Z}\Phi(L)\) | yes |
|---|---|
| \(L = \mathbb{Z}\Phi_S(L)\) for | \(S = \{2\}\) |
| Number of roots | 288 |
| \(R(L) = \mathbb{Z}\Phi(L)\) | \(\cong E_{8} \oplus D_{4}\) |
| Rank of \(R(L)\) | 12 |
| \([L : R(L)]\) | 1 |
| \(R(L)\) is primitive in \(L\) | yes |
Components of \(\Phi(L)\)
| Component | Type | Values of \(b(r, r)\) | Number of roots | \(\mathbb{Z}\Phi_c\) |
|---|---|---|---|---|
| \(\Phi_{1}\) | \(E_{8}\) | \(\{2\}\) | 240 | \(\cong E_{8}\) |
| \(\Phi_{2}\) | \(F_{4}\) | \(\{2, 4\}\) | 48 | \(\cong D_{4}\) |
Simple roots
\(\alpha^{(c)}_j\) is simple root \(j\) of the component \(\Phi_c\).
| Simple root | Coordinates | \(b(r, r)\) |
|---|---|---|
| \(\alpha^{(1)}_{1}\) | \(e_{1}\) | \(2\) |
| \(\alpha^{(1)}_{2}\) | \(e_{2}\) | \(2\) |
| \(\alpha^{(1)}_{3}\) | \(e_{3}\) | \(2\) |
| \(\alpha^{(1)}_{4}\) | \(e_{4}\) | \(2\) |
| \(\alpha^{(1)}_{5}\) | \(e_{5}\) | \(2\) |
| \(\alpha^{(1)}_{6}\) | \(e_{6}\) | \(2\) |
| \(\alpha^{(1)}_{7}\) | \(e_{7}\) | \(2\) |
| \(\alpha^{(1)}_{8}\) | \(e_{8}\) | \(2\) |
| \(\alpha^{(2)}_{1}\) | \(e_{9} - e_{11}\) | \(4\) |
| \(\alpha^{(2)}_{2}\) | \(e_{11} - e_{12}\) | \(4\) |
| \(\alpha^{(2)}_{3}\) | \(e_{12}\) | \(2\) |
| \(\alpha^{(2)}_{4}\) | \(e_{10}\) | \(2\) |