U(5) + (-4 -2; -2 -6) · Lattice Database

Rank	4
Signature \((n_+, n_-)\)	(1, 3)
Determinant	\(-500\)

Rank

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

(1, 3)

\(-500\)

Gram tensor

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

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

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

[[0, -5, 0, 0], [-5, 0, 0, 0], [0, 0, -4, -2], [0, 0, -2, -6]]

Integral lattice

Parity	even
Discriminant group	\(\mathbb{Z}/5 \oplus (\mathbb{Z}/10)^{2}\)
Unimodular	no
Integral overlattices \(L \subseteq M \subseteq L^*\)	14
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)	\(\{2, 5\}\)
Character of the discriminant \(\chi_D\), \(D = (-1)^{2} \det L\)	\(\chi_{-20} = \left(\frac{-20}{\cdot}\right)\)
Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)	\(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 3)\, L^\Sigma(s - 2, \chi_{-20})}{L^\Sigma(s - 1, \chi_{-20})}\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)	\(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 3)}{L^\Sigma(s - 1, \chi_{-20})}\)

Parity

even

Discriminant group

\(\mathbb{Z}/5 \oplus (\mathbb{Z}/10)^{2}\)

Unimodular

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

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

\(\{2, 5\}\)

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

\(\chi_{-20} = \left(\frac{-20}{\cdot}\right)\)

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

\(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 3)\, L^\Sigma(s - 2, \chi_{-20})}{L^\Sigma(s - 1, \chi_{-20})}\)

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

\(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 3)}{L^\Sigma(s - 1, \chi_{-20})}\)

Isotropic	yes

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)	yes
\(L = \mathbb{Z}\Phi_S(L)\) for	\(S = \{-10, -4, 10\}\)
\(R(L) = \mathbb{Z}\Phi(L)\)	\(L\)
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

\(S = \{-10, -4, 10\}\)

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

\(L\)

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	\(b(r, r)\)
\(r_{1}\)	\(e_{3}\)	\(-4\)
\(r_{2}\)	\(e_{1} + e_{2}\)	\(-10\)
\(r_{3}\)	\(2e_{1} - e_{2} + 2e_{3} + e_{4}\)	\(-10\)
\(r_{4}\)	\(2e_{1} - 2e_{2} + 2e_{3} + e_{4}\)	\(10\)

Notes

Let \(\Lambda = \mathrm{II}_{3,19}\) be the K3 lattice and \(G \subset O(\Lambda)\) a finite group of symplectic isometries. Row 32 of Table 10.3 of Hashimoto gives the invariant lattice \(\Lambda^G\) for the group \(\mathfrak{G}_{32} \cong \mathrm{Hol}(C_5)\) of Table 10.2, and this record is \(\Lambda^G(-1)\); \(\Lambda^G\) has signature \((3, 1)\) and the corpus records the sign with \(n_+ \leq n_-\).

Roots: \(e_{3}\) has \(b(r, r) = -4\) and \(b(r, L) = 2\mathbb{Z}\), and \(-4\) divides \(4\). \(e_{1} + e_{2}\) and \(2 e_{1} - e_{2} + 2 e_{3} + e_{4}\) have \(b(r, r) = -10\) and \(b(r, L) = 5\mathbb{Z}\), and \(-10\) divides \(10\). \(2 e_{1} - 2 e_{2} + 2 e_{3} + e_{4}\) has \(b(r, r) = 10\) and \(b(r, L) = 10\mathbb{Z}\), and \(10\) divides \(20\). These roots generate \(L\), so \(L = \mathbb{Z}\Phi(L)\).

Related lattices

\(U(5) \oplus \begin{pmatrix} -4 & -2 \\ -2 & -6 \end{pmatrix}\)

Invariants

Gram tensor

Integral lattice

Indefinite lattice

Roots

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

Notes

Morphisms

References