Tag 0046
\(\mathrm{I}_{2,2}\)
I_{2,2}
Invariants
| Rank | 4 |
|---|---|
| Signature \((n_+, n_-)\) | (2, 2) |
| Determinant | \(1\) |
Gram tensor
The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{4}\).
In this basis, \(L = {}\)\(\mathbb{Z}\{e_{1}\}\) \(\oplus\) \(\mathbb{Z}\{e_{2}\}\) \(\oplus\) \(\mathbb{Z}\{e_{3}\}\) \(\oplus\) \(\mathbb{Z}\{e_{4}\}\), an orthogonal sum.
| \(e_{1}\) | \(e_{2}\) | \(e_{3}\) | \(e_{4}\) | |
|---|---|---|---|---|
| \(e_{1}\) | 1 | 0 | 0 | 0 |
| \(e_{2}\) | 0 | 1 | 0 | 0 |
| \(e_{3}\) | 0 | 0 | -1 | 0 |
| \(e_{4}\) | 0 | 0 | 0 | -1 |
[[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, -1, 0], [0, 0, 0, -1]]
Integral lattice
| Parity | odd |
|---|---|
| Discriminant group | \(0\) |
| Unimodular | yes |
| Integral overlattices \(L \subseteq M \subseteq L^*\) | 1 |
| Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\) | \(\{2\}\) |
| Character of the discriminant \(\chi_D\), \(D = (-1)^{2} \det L\) | trivial |
| Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\) | \(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 3)\, \zeta^\Sigma(s - 2)}{\zeta^\Sigma(s - 1)}\) |
| Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\) | \(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 3)}{\zeta^\Sigma(s - 1)}\) |
| Genus symbol | \(\mathrm{I}_{2,2}\) |
| Class number of the genus | 1 |
Indefinite lattice
| Isotropic | yes |
|---|
Roots
| Root lattice: \(L = \mathbb{Z}\Phi(L)\) | yes |
|---|---|
| \(L = \mathbb{Z}\Phi_S(L)\) for | \(S = \{-1, 1\}\) |
| \(R(L) = \mathbb{Z}\Phi(L)\) | \(L\) |
| Rank of \(R(L)\) | 4 |
| \([L : R(L)]\) | 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 | \(b(r, r)\) |
|---|---|---|
| \(r_{1}\) | \(e_{1}\) | \(1\) |
| \(r_{2}\) | \(e_{2}\) | \(1\) |
| \(r_{3}\) | \(e_{3}\) | \(-1\) |
| \(r_{4}\) | \(e_{4}\) | \(-1\) |