U + <-2> + <2> + <-4> · Lattice Database

Rank	5
Signature \((n_+, n_-)\)	(2, 3)
Determinant	\(-16\)

Rank

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

(2, 3)

\(-16\)

Gram tensor

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

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

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

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

Integral lattice

Parity	even
Discriminant group	\((\mathbb{Z}/2)^{2} \oplus \mathbb{Z}/4\)
Unimodular	no
Integral overlattices \(L \subseteq M \subseteq L^*\)	5
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 \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)	\(\zeta^\Sigma(X_n, s) = \zeta^\Sigma(s - 4)\, L^\Sigma(s - 2, \chi_{-16n})\)

Parity

even

Discriminant group

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

Unimodular

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

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

\(\zeta^\Sigma(X_n, s) = \zeta^\Sigma(s - 4)\, L^\Sigma(s - 2, \chi_{-16n})\)

Isotropic	yes

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)	yes
\(L = \mathbb{Z}\Phi_S(L)\) for	\(S = \{-4, -2, 2\}\)
\(R(L) = \mathbb{Z}\Phi(L)\)	\(L\)
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 \(L\).

Root lattice: \(L = \mathbb{Z}\Phi(L)\)

yes

\(L = \mathbb{Z}\Phi_S(L)\) for

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

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

\(L\)

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

Notes

Let \(\Lambda = \mathrm{II}_{3,19}\) be the K3 lattice and \(G \subset O(\Lambda)\) a finite group of symplectic isometries. Row 39 of Table 10.3 of Hashimoto gives the invariant lattice \(\Lambda^G\) for the groups \(\mathfrak{G}_{39} \cong 2^4 C_2\), \(\mathfrak{G}_{23} \cong \Gamma_2 c_1\) 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

\(U \oplus \langle -2 \rangle \oplus \langle 2 \rangle \oplus \langle -4 \rangle\)

Invariants

Gram tensor

Integral lattice

Indefinite lattice

Roots

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

Notes

Morphisms

References