U + <2>^2 + <-2>^2 · Lattice Database

Rank	6
Signature \((n_+, n_-)\)	(3, 3)
Determinant	\(-16\)

Rank

Signature \((n_+, n_-)\)

(3, 3)

\(-16\)

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

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

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

Integral lattice

Parity	even
Discriminant group	\((\mathbb{Z}/2)^{4}\)
Unimodular	no
Integral overlattices \(L \subseteq M \subseteq L^*\)	11
Nikulin invariants \((r, a, \delta)\)	\((6, 4, 1)\)
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 \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)	\(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 5)}{\zeta^\Sigma(s - 2)}\)
Hyperbolic index \(\max\{n : L \cong U^n \oplus L'\}\)	\(\geq 1\), by an embedding of \(U^{1}\) in the morphism files

Parity

even

Discriminant group

\((\mathbb{Z}/2)^{4}\)

Unimodular

Integral overlattices \(L \subseteq M \subseteq L^*\)

Nikulin invariants \((r, a, \delta)\)

\((6, 4, 1)\)

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 \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)

\(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 5)}{\zeta^\Sigma(s - 2)}\)

Hyperbolic index \(\max\{n : L \cong U^n \oplus L'\}\)

\(\geq 1\), by an embedding of \(U^{1}\) in the morphism files

Isotropic	yes

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 \(R(L)\)	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 \(L\).

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 \(R(L)\)

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

\(R(L)\) 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	\(b(r, r)\)
\(r_{1}\)	\(e_{3}\)	\(2\)
\(r_{2}\)	\(e_{4}\)	\(2\)
\(r_{3}\)	\(e_{5}\)	\(-2\)
\(r_{4}\)	\(e_{6}\)	\(-2\)
\(r_{5}\)	\(e_{1} + e_{2}\)	\(2\)
\(r_{6}\)	\(e_{1} + e_{3}\)	\(2\)

Notes

Let \(\Lambda = \mathrm{II}_{3,19}\) be the K3 lattice and \(G \subset O(\Lambda)\) a finite group of symplectic isometries. Row 22 of Table 10.3 of Hashimoto gives the invariant lattice \(\Lambda^G\) for the groups \(\mathfrak{G}_{22} \cong C_2 \times D_8\), \(\mathfrak{G}_{11} \cong C_2 \times C_4\) of Table 10.2, and this record is the lattice \(M\) with \(\Lambda^G = M(2)\); the corpus records \(M\) and not its twist.

Related lattices

\(U \oplus \langle 2 \rangle^{2} \oplus \langle -2 \rangle^{2}\)

Invariants

Gram tensor

Integral lattice

Indefinite lattice

Roots

Roots that generate \(\mathbb{Z}\Phi(L)\)

Notes

Morphisms

References