Tag 0039

\(U \oplus A_{2}(-1)\)

U + A2(-1)

Invariants

Rank4
Signature \((n_+, n_-)\)(1, 3)
Determinant\(-3\)

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, \(L = {}\)\(\mathbb{Z}\{e_{1}, e_{2}\}\) \(\oplus\) \(\mathbb{Z}\{e_{3}, e_{4}\}\), an orthogonal sum.

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)
\(e_{1}\)0100
\(e_{2}\)1000
\(e_{3}\)00-21
\(e_{4}\)001-2
 
[[0, 1, 0, 0], [1, 0, 0, 0], [0, 0, -2, 1], [0, 0, 1, -2]]

Integral lattice

Parityeven
Discriminant group\(\mathbb{Z}/3\)
Unimodularno
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 \(\chi_D\), \(D = (-1)^{2} \det L\)\(\chi_{-3} = \left(\frac{-3}{\cdot}\right)\)
Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)\(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 3)\, L^\Sigma(s - 2, \chi_{-3})}{L^\Sigma(s - 1, \chi_{-3})}\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)\(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 3)}{L^\Sigma(s - 1, \chi_{-3})}\)
Genus symbol\(\mathrm{II}_{1,3}\ \bigl(2\colon 1^{-4};\ 3\colon 1^{-3}\,3^{1}\bigr)\)
Class number of the genus1
Hyperbolic index \(\max\{n : L \cong U^n \oplus L'\}\)\(\geq 1\), by an embedding of \(U^{1}\) in the morphism files

Indefinite lattice

Isotropicyes

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)yes
\(L = \mathbb{Z}\Phi_S(L)\) for\(S = \{-2, 2\}\)
\(R(L) = \mathbb{Z}\Phi(L)\)\(L\)
Rank of \(R(L)\)4
\([L : R(L)]\)1
\(R(L)\) is primitive in \(L\)yes
ProofEach \(r_i\) 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 \(L\).

RootCoordinates\(b(r, r)\)
\(r_{1}\)\(e_{3}\)\(-2\)
\(r_{2}\)\(e_{4}\)\(-2\)
\(r_{3}\)\(e_{1} + e_{2}\)\(2\)
\(r_{4}\)\(e_{1} + e_{3}\)\(-2\)

Morphisms