Tag 0053

\(U \oplus A_{3}(-1)\)

U + A3(-1)

Invariants

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{5}\).

In this basis, \(L = {}\)\(\mathbb{Z}\{e_{1}, e_{2}\}\) \(\oplus\) \(\mathbb{Z}\{e_{3}, e_{4}, e_{5}\}\), an orthogonal sum.

[[0, 1, 0, 0, 0], [1, 0, 0, 0, 0], [0, 0, -2, 1, 0], [0, 0, 1, -2, 1], [0, 0, 0, 1, -2]]

Parity	even
Discriminant group	\(\mathbb{Z}/4\)
Unimodular	no
Integral overlattices \(L \subseteq M \subseteq L^*\)	2
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)	\(\{2\}\)
Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)	\(\zeta^\Sigma(X_0, s) = \zeta^\Sigma(s - 4)\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)	\(\zeta^\Sigma(X_n, s) = \zeta^\Sigma(s - 4)\, L^\Sigma(s - 2, \chi_{4n})\)
Genus symbol	\(\mathrm{II}_{1,4}\ \bigl(2\colon 1^{-4}:[4^{-1}]_{5}\bigr)\)
Class number of the genus	1
Hyperbolic index \(\max\{n : L \cong U^n \oplus L'\}\)	\(\geq 1\), by an embedding of \(U^{1}\) in the morphism files

Isotropic	yes

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 \(R(L)\)	5
\([L : R(L)]\)	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 \(L\).

The \(r_i\) are a basis of \(L\).