U + U(5)^2 · Lattice Database

Rank	6
Signature \((n_+, n_-)\)	(3, 3)
Determinant	\(-625\)

Rank

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

(3, 3)

\(-625\)

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, \(L = {}\)\(\mathbb{Z}\{e_{1}, e_{2}\}\) \(\oplus\) \(\mathbb{Z}\{e_{3}, e_{4}\}\) \(\oplus\) \(\mathbb{Z}\{e_{5}, e_{6}\}\), 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	0	5	0	0
\(e_{4}\)	0	0	5	0	0	0
\(e_{5}\)	0	0	0	0	0	5
\(e_{6}\)	0	0	0	0	5	0

[[0, 1, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [0, 0, 0, 5, 0, 0], [0, 0, 5, 0, 0, 0], [0, 0, 0, 0, 0, 5], [0, 0, 0, 0, 5, 0]]

Integral lattice

Parity	even
Discriminant group	\((\mathbb{Z}/5)^{4}\)
Unimodular	no
Integral overlattices \(L \subseteq M \subseteq L^*\)	49
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)	\(\{2, 5\}\)
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

\((\mathbb{Z}/5)^{4}\)

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 \(\chi_D\), \(D = (-1)^{3} \det L\)

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	\(S = \{2, 10\}\)
\(R(L) = \mathbb{Z}\Phi(L)\)	\(L\)
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

\(S = \{2, 10\}\)

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

\(L\)

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	\(b(r, r)\)
\(r_{1}\)	\(e_{1} + e_{2}\)	\(2\)
\(r_{2}\)	\(e_{3} + e_{4}\)	\(10\)
\(r_{3}\)	\(e_{5} + e_{6}\)	\(10\)
\(r_{4}\)	\(e_{1} + e_{2} + e_{3}\)	\(2\)
\(r_{5}\)	\(e_{1} + e_{2} + e_{5}\)	\(2\)
\(r_{6}\)	\(3e_{1} + 2e_{2} + e_{3} - e_{4}\)	\(2\)

Notes

Let \(\Lambda = \mathrm{II}_{3,19}\) be the K3 lattice and \(G \subset O(\Lambda)\) a finite group of symplectic isometries. Row 16 of Table 10.3 of Hashimoto gives the invariant lattice \(\Lambda^G\) for the groups \(\mathfrak{G}_{16} \cong D_{10}\), \(\mathfrak{G}_{5} \cong C_5\) of Table 10.2, and this record is \(\Lambda^G\).

Roots: \(e_{1} + e_{2}\), \(e_{1} + e_{2} + e_{3}\), \(e_{1} + e_{2} + e_{5}\) and \(3 e_{1} + 2 e_{2} + e_{3} - e_{4}\) have \(b(r, r) = 2\) and \(b(r, L) = \mathbb{Z}\), and \(2\) divides \(2\). \(e_{3} + e_{4}\) and \(e_{5} + e_{6}\) have \(b(r, r) = 10\) and \(b(r, L) = 5\mathbb{Z}\), and \(10\) divides \(10\). These roots generate \(L\), so \(L = \mathbb{Z}\Phi(L)\).

Related lattices

\(U \oplus U(5)^{2}\)

Invariants

Gram tensor

Integral lattice

Indefinite lattice

Roots

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

Notes

Morphisms

References