Tag 0012
\(A_{2}\)
A2
Invariants
| Rank | 2 |
|---|---|
| Signature \((n_+, n_-)\) | (2, 0) |
| Determinant | \(3\) |
Gram tensor
The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, e_{2}\).
| \(e_{1}\) | \(e_{2}\) | |
|---|---|---|
| \(e_{1}\) | 2 | -1 |
| \(e_{2}\) | -1 | 2 |
[[2, -1], [-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 \(\chi_D\), \(D = (-1)^{1} \det L\) | \(\chi_{-3} = \left(\frac{-3}{\cdot}\right)\) |
| Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\) | \(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 1)\, L^\Sigma(s - 1, \chi_{-3})}{L^\Sigma(s, \chi_{-3})}\) |
| Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\) | \(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 1)}{L^\Sigma(s, \chi_{-3})}\) |
| Genus symbol | \(\mathrm{II}_{2,0}\ \bigl(2\colon 1^{-2};\ 3\colon 1^{-1}\,3^{-1}\bigr)\) |
| Class number of the genus | 1 |
Definite lattice
| Minimum of \(b(x, x)\), \(x \neq 0\) | \(2\) |
|---|---|
| Kissing number | 6 |
| Order of the isometry group | \(12 = 2^{2} \cdot 3\) |
Theta series
\[ \sum_{x \in L} q^{b(x, x)} = 1 + 6q^{2} + 6q^{6} + 6q^{8} + O(q^{13}) \]Roots
| Root lattice: \(L = \mathbb{Z}\Phi(L)\) | yes |
|---|---|
| \(L = \mathbb{Z}\Phi_S(L)\) for | \(S = \{2\}\) |
| Number of roots | 12 |
| \(R(L) = \mathbb{Z}\Phi(L)\) | \(\cong A_{2}\) |
| Rank of \(R(L)\) | 2 |
| \([L : R(L)]\) | 1 |
| \(R(L)\) is primitive in \(L\) | yes |
Components of \(\Phi(L)\)
| Component | Type | Values of \(b(r, r)\) | Number of roots | \(\mathbb{Z}\Phi_c\) |
|---|---|---|---|---|
| \(\Phi_{1}\) | \(G_{2}\) | \(\{2, 6\}\) | 12 | \(\cong A_{2}\) |
Simple roots
| Simple root | Coordinates | \(b(r, r)\) |
|---|---|---|
| \(\alpha_{1}\) | \(e_{2}\) | \(2\) |
| \(\alpha_{2}\) | \(e_{1} - e_{2}\) | \(6\) |