Tag 0065

\(\mathrm{I}_{1,5}\)

I_{1,5}

Invariants

Rank	6
Signature \((n_+, n_-)\)	(1, 5)
Determinant	\(-1\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{6}\).

In this basis, \(L = {}\)\(\mathbb{Z}\{e_{1}\}\) \(\oplus\) \(\mathbb{Z}\{e_{2}\}\) \(\oplus\) \(\mathbb{Z}\{e_{3}\}\) \(\oplus\) \(\mathbb{Z}\{e_{4}\}\) \(\oplus\) \(\mathbb{Z}\{e_{5}\}\) \(\oplus\) \(\mathbb{Z}\{e_{6}\}\), an orthogonal sum.

	\(e_{1}\)	\(e_{2}\)	\(e_{3}\)	\(e_{4}\)	\(e_{5}\)	\(e_{6}\)
\(e_{1}\)	1	0	0	0	0	0
\(e_{2}\)	0	-1	0	0	0	0
\(e_{3}\)	0	0	-1	0	0	0
\(e_{4}\)	0	0	0	-1	0	0
\(e_{5}\)	0	0	0	0	-1	0
\(e_{6}\)	0	0	0	0	0	-1

[[1, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0], [0, 0, 0, -1, 0, 0], [0, 0, 0, 0, -1, 0], [0, 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 \(\chi_D\), \(D = (-1)^{3} \det L\)	trivial
Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)	\(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 5)\, \zeta^\Sigma(s - 3)}{\zeta^\Sigma(s - 2)}\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)	\(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 5)}{\zeta^\Sigma(s - 2)}\)
Genus symbol	\(\mathrm{I}_{1,5}\)
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 \(R(L)\)	6
\([L : R(L)]\)	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	\(b(r, r)\)
\(r_{1}\)	\(e_{1}\)	\(1\)
\(r_{2}\)	\(e_{2}\)	\(-1\)
\(r_{3}\)	\(e_{3}\)	\(-1\)
\(r_{4}\)	\(e_{4}\)	\(-1\)
\(r_{5}\)	\(e_{5}\)	\(-1\)
\(r_{6}\)	\(e_{6}\)	\(-1\)

Morphisms