Tag 0269
\(\langle -1 \rangle^{3} \oplus \langle 7 \rangle\)
<-1>^3 + <7>
Invariants
| Rank | 4 |
|---|---|
| Signature \((n_+, n_-)\) | (1, 3) |
| Determinant | \(-7\) |
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 | 7 |
[[-1, 0, 0, 0], [0, -1, 0, 0], [0, 0, -1, 0], [0, 0, 0, 7]]
Integral lattice
| Parity | odd |
|---|---|
| Discriminant group | \(\mathbb{Z}/7\) |
| Unimodular | no |
| Integral overlattices \(L \subseteq M \subseteq L^*\) | 1 |
| Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\) | \(\{2, 7\}\) |
| Character of the discriminant \(\chi_D\), \(D = (-1)^{2} \det L\) | \(\chi_{-7} = \left(\frac{-7}{\cdot}\right)\) |
| Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\) | \(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 3)\, L^\Sigma(s - 2, \chi_{-7})}{L^\Sigma(s - 1, \chi_{-7})}\) |
| Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\) | \(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 3)}{L^\Sigma(s - 1, \chi_{-7})}\) |
Indefinite lattice
| Isotropic | no |
|---|
Roots
| Root lattice: \(L = \mathbb{Z}\Phi(L)\) | yes |
|---|---|
| \(L = \mathbb{Z}\Phi_S(L)\) for | \(S = \{-1, 7\}\) |
| \(R(L) = \mathbb{Z}\Phi(L)\) | \(L\) |
| 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 | \(b(r, r)\) |
|---|---|---|
| \(r_{1}\) | \(e_{1}\) | \(-1\) |
| \(r_{2}\) | \(e_{2}\) | \(-1\) |
| \(r_{3}\) | \(e_{3}\) | \(-1\) |
| \(r_{4}\) | \(e_{4}\) | \(7\) |
Notes
\(L\) is anisotropic: \(b(x, x) = 0\) means \(x_1^2 + x_2^2 + x_3^2 = 7x_4^2\). For \(x_4\) odd the right side is \(7\) modulo \(8\), which is not a sum of three squares modulo 8. For \(x_4\) even the left side is \(0\) modulo \(4\), so \(x_1\), \(x_2\), \(x_3\) are even, and descent gives \(x = 0\). It is an odd lattice of signature \((1, 3)\).