(-2 0 -1 0; 0 -2 -1 0; -1 -1 -2 -2; 0 0 -2 0)

Rank	4
Signature \((n_+, n_-)\)	(1, 3)
Determinant	\(-16\)

Rank

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

(1, 3)

\(-16\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{4}\).

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

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

Integral lattice

Parity	even
Discriminant group	\((\mathbb{Z}/4)^{2}\)
Unimodular	no
Integral overlattices \(L \subseteq M \subseteq L^*\)	5
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)	\(\{2\}\)
Character of the discriminant \(\chi_D\), \(D = (-1)^{2} \det L\)	\(\chi_{-4} = \left(\frac{-4}{\cdot}\right)\)
Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)	\(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 3)\, L^\Sigma(s - 2, \chi_{-4})}{L^\Sigma(s - 1, \chi_{-4})}\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)	\(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 3)}{L^\Sigma(s - 1, \chi_{-4})}\)

Parity

even

Discriminant group

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

Unimodular

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

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

\(\{2\}\)

Character of the discriminant \(\chi_D\), \(D = (-1)^{2} \det L\)

\(\chi_{-4} = \left(\frac{-4}{\cdot}\right)\)

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

\(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 3)\, L^\Sigma(s - 2, \chi_{-4})}{L^\Sigma(s - 1, \chi_{-4})}\)

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

\(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 3)}{L^\Sigma(s - 1, \chi_{-4})}\)

Isotropic	yes

Roots

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

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

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

\(L\)

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

Notes

Let \(\Lambda = \mathrm{II}_{3,19}\) be the K3 lattice and \(G \subset O(\Lambda)\) a finite group of symplectic isometries. Row 75 of Table 10.3 of Hashimoto gives the invariant lattice \(\Lambda^G\) for the groups \(\mathfrak{G}_{75} \cong 4^2 \mathfrak{A}_4\), \(\mathfrak{G}_{25} \cong C_4^2\), \(\mathfrak{G}_{27} \cong C_2 \times Q_8\), \(\mathfrak{G}_{42} \cong \Gamma_4 c_2\), \(\mathfrak{G}_{50} \cong 4^2 C_3\), \(\mathfrak{G}_{57} \cong \Gamma_{13} a_1\) 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

\(\begin{pmatrix} -2 & 0 & -1 & 0 \\ 0 & -2 & -1 & 0 \\ -1 & -1 & -2 & -2 \\ 0 & 0 & -2 & 0 \end{pmatrix}\)

Invariants

Gram tensor

Integral lattice

Indefinite lattice

Roots

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

Notes

Morphisms

References