Tag 0021

\(\langle 1 \rangle \oplus \langle -2 \rangle\)

<1> + <-2>

Invariants

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, e_{2}\).

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

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

Parity	odd
Discriminant group	\(\mathbb{Z}/2\)
Unimodular	no
Integral overlattices \(L \subseteq M \subseteq L^*\)	1
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)	\(\{2\}\)
Character of the discriminant \(\chi_D\), \(D = (-1)^{1} \det L\)	\(\chi_{8} = \left(\frac{8}{\cdot}\right)\)
Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)	\(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 1)\, L^\Sigma(s - 1, \chi_{8})}{L^\Sigma(s, \chi_{8})}\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)	\(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 1)}{L^\Sigma(s, \chi_{8})}\)
Genus symbol	\(\mathrm{I}_{1,1}\ \bigl(2\colon [1^{1}\,2^{1}]_{0}\bigr)\)
Class number of the genus	1

Isotropic	no

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

Root	Coordinates	\(b(r, r)\)
\(r_{1}\)	\(e_{1}\)	\(1\)
\(r_{2}\)	\(e_{2}\)	\(-2\)