U(3) + A2(2) + A2(-1)^2 · Lattice Database

Rank	8
Signature \((n_+, n_-)\)	(3, 5)
Determinant	\(-972\)

Rank

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

(3, 5)

\(-972\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{8}\).

In this basis, \(L = {}\)\(\mathbb{Z}\{e_{1}, e_{2}\}\) \(\oplus\) \(\mathbb{Z}\{e_{3}, e_{4}\}\) \(\oplus\) \(\mathbb{Z}\{e_{5}, e_{6}\}\) \(\oplus\) \(\mathbb{Z}\{e_{7}, e_{8}\}\), an orthogonal sum.

	\(e_{1}\)	\(e_{2}\)	\(e_{3}\)	\(e_{4}\)	\(e_{5}\)	\(e_{6}\)	\(e_{7}\)	\(e_{8}\)
\(e_{1}\)	0	3	0	0	0	0	0	0
\(e_{2}\)	3	0	0	0	0	0	0	0
\(e_{3}\)	0	0	4	-2	0	0	0	0
\(e_{4}\)	0	0	-2	4	0	0	0	0
\(e_{5}\)	0	0	0	0	-2	1	0	0
\(e_{6}\)	0	0	0	0	1	-2	0	0
\(e_{7}\)	0	0	0	0	0	0	-2	1
\(e_{8}\)	0	0	0	0	0	0	1	-2

[[0, 3, 0, 0, 0, 0, 0, 0], [3, 0, 0, 0, 0, 0, 0, 0], [0, 0, 4, -2, 0, 0, 0, 0], [0, 0, -2, 4, 0, 0, 0, 0], [0, 0, 0, 0, -2, 1, 0, 0], [0, 0, 0, 0, 1, -2, 0, 0], [0, 0, 0, 0, 0, 0, -2, 1], [0, 0, 0, 0, 0, 0, 1, -2]]

Integral lattice

Parity	even
Discriminant group	\((\mathbb{Z}/3)^{3} \oplus (\mathbb{Z}/6)^{2}\)
Unimodular	no
Integral overlattices \(L \subseteq M \subseteq L^*\)	324
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)	\(\{2, 3\}\)
Character of the discriminant \(\chi_D\), \(D = (-1)^{4} \det L\)	\(\chi_{-3} = \left(\frac{-3}{\cdot}\right)\)
Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)	\(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 7)\, L^\Sigma(s - 4, \chi_{-3})}{L^\Sigma(s - 3, \chi_{-3})}\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)	\(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 7)}{L^\Sigma(s - 3, \chi_{-3})}\)

Parity

even

Discriminant group

\((\mathbb{Z}/3)^{3} \oplus (\mathbb{Z}/6)^{2}\)

Unimodular

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

324

Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)

\(\{2, 3\}\)

Character of the discriminant \(\chi_D\), \(D = (-1)^{4} \det L\)

\(\chi_{-3} = \left(\frac{-3}{\cdot}\right)\)

Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)

\(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 7)\, L^\Sigma(s - 4, \chi_{-3})}{L^\Sigma(s - 3, \chi_{-3})}\)

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

\(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 7)}{L^\Sigma(s - 3, \chi_{-3})}\)

Isotropic	yes

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)	yes
\(L = \mathbb{Z}\Phi_S(L)\) for	\(S = \{-2, 4, 6\}\)
\(R(L) = \mathbb{Z}\Phi(L)\)	\(L\)
Rank of \(R(L)\)	8
\([L : R(L)]\)	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

\(S = \{-2, 4, 6\}\)

\(R(L) = \mathbb{Z}\Phi(L)\)

\(L\)

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	\(b(r, r)\)
\(r_{1}\)	\(e_{3}\)	\(4\)
\(r_{2}\)	\(e_{4}\)	\(4\)
\(r_{3}\)	\(e_{5}\)	\(-2\)
\(r_{4}\)	\(e_{6}\)	\(-2\)
\(r_{5}\)	\(e_{7}\)	\(-2\)
\(r_{6}\)	\(e_{8}\)	\(-2\)
\(r_{7}\)	\(e_{1} + e_{2}\)	\(6\)
\(r_{8}\)	\(e_{1} + e_{5}\)	\(-2\)

Let \(\Lambda = \mathrm{II}_{3,19}\) be the K3 lattice and \(G \subset O(\Lambda)\) a finite group of symplectic isometries. Row 6 of Table 10.3 of Hashimoto gives the invariant lattice \(\Lambda^G\) for the group \(\mathfrak{G}_{6} \cong D_6\) of Table 10.2, and this record is \(\Lambda^G\).

Related lattices

\(U(3) \oplus A_{2}(2) \oplus A_{2}(-1)^{2}\)

Invariants

Gram tensor

Integral lattice

Indefinite lattice

Roots

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

Notes

Morphisms

References