Tag 0064
\(E_{6}\)
E6
Invariants
| Rank | 6 |
|---|---|
| Signature \((n_+, n_-)\) | (6, 0) |
| Determinant | \(3\) |
Gram tensor
The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{6}\).
| \(e_{1}\) | \(e_{2}\) | \(e_{3}\) | \(e_{4}\) | \(e_{5}\) | \(e_{6}\) | |
|---|---|---|---|---|---|---|
| \(e_{1}\) | 2 | 0 | -1 | 0 | 0 | 0 |
| \(e_{2}\) | 0 | 2 | 0 | -1 | 0 | 0 |
| \(e_{3}\) | -1 | 0 | 2 | -1 | 0 | 0 |
| \(e_{4}\) | 0 | -1 | -1 | 2 | -1 | 0 |
| \(e_{5}\) | 0 | 0 | 0 | -1 | 2 | -1 |
| \(e_{6}\) | 0 | 0 | 0 | 0 | -1 | 2 |
[[2, 0, -1, 0, 0, 0], [0, 2, 0, -1, 0, 0], [-1, 0, 2, -1, 0, 0], [0, -1, -1, 2, -1, 0], [0, 0, 0, -1, 2, -1], [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 \(\chi_D\), \(D = (-1)^{3} \det L\) | \(\chi_{-3} = \left(\frac{-3}{\cdot}\right)\) |
| Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\) | \(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 5)\, L^\Sigma(s - 3, \chi_{-3})}{L^\Sigma(s - 2, \chi_{-3})}\) |
| Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\) | \(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 5)}{L^\Sigma(s - 2, \chi_{-3})}\) |
| Genus symbol | \(\mathrm{II}_{6,0}\ \bigl(2\colon 1^{-6};\ 3\colon 1^{5}\,3^{1}\bigr)\) |
| Class number of the genus | 1 |
Definite lattice
| Minimum of \(b(x, x)\), \(x \neq 0\) | \(2\) |
|---|---|
| Kissing number | 72 |
| Order of the isometry group | \(103680 = 2^{8} \cdot 3^{4} \cdot 5\) |
Theta series
\[ \sum_{x \in L} q^{b(x, x)} = 1 + 72q^{2} + 270q^{4} + 720q^{6} + 936q^{8} + O(q^{9}) \]Roots
| Root lattice: \(L = \mathbb{Z}\Phi(L)\) | yes |
|---|---|
| \(L = \mathbb{Z}\Phi_S(L)\) for | \(S = \{2\}\) |
| Number of roots | 72 |
| \(R(L) = \mathbb{Z}\Phi(L)\) | \(\cong E_{6}\) |
| Rank of \(R(L)\) | 6 |
| \([L : R(L)]\) | 1 |
| \(R(L)\) is primitive in \(L\) | yes |
Components of \(\Phi(L)\)
| Component | Type | Values of \(b(r, r)\) | Number of roots | \(\mathbb{Z}\Phi_c\) |
|---|---|---|---|---|
| \(\Phi_{1}\) | \(E_{6}\) | \(\{2\}\) | 72 | \(\cong E_{6}\) |
Simple roots
| Simple root | Coordinates | \(b(r, r)\) |
|---|---|---|
| \(\alpha_{1}\) | \(e_{6}\) | \(2\) |
| \(\alpha_{2}\) | \(e_{2}\) | \(2\) |
| \(\alpha_{3}\) | \(e_{5}\) | \(2\) |
| \(\alpha_{4}\) | \(e_{4}\) | \(2\) |
| \(\alpha_{5}\) | \(e_{3}\) | \(2\) |
| \(\alpha_{6}\) | \(e_{1}\) | \(2\) |