U^3 + <-8> · Lattice Database

Rank	7
Signature \((n_+, n_-)\)	(3, 4)
Determinant	\(8\)

Rank

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

(3, 4)

\(8\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{7}\).

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

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

[[0, 1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, -8]]

Integral lattice

Parity	even
Discriminant group	\(\mathbb{Z}/8\)
Unimodular	no
Integral overlattices \(L \subseteq M \subseteq L^*\)	2
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)	\(\{2\}\)
Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)	\(\zeta^\Sigma(X_0, s) = \zeta^\Sigma(s - 6)\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)	\(\zeta^\Sigma(X_n, s) = \zeta^\Sigma(s - 6)\, L^\Sigma(s - 3, \chi_{-8n})\)
Hyperbolic index \(\max\{n : L \cong U^n \oplus L'\}\)	\(\geq 3\), by an embedding of \(U^{3}\) in the morphism files

Parity

even

Discriminant group

\(\mathbb{Z}/8\)

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 \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)

\(\zeta^\Sigma(X_0, s) = \zeta^\Sigma(s - 6)\)

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

\(\zeta^\Sigma(X_n, s) = \zeta^\Sigma(s - 6)\, L^\Sigma(s - 3, \chi_{-8n})\)

Hyperbolic index \(\max\{n : L \cong U^n \oplus L'\}\)

\(\geq 3\), by an embedding of \(U^{3}\) in the morphism files

Isotropic	yes

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)	yes
\(L = \mathbb{Z}\Phi_S(L)\) for	\(S = \{-8, 2\}\)
\(R(L) = \mathbb{Z}\Phi(L)\)	\(L\)
Rank of \(R(L)\)	7
\([L : R(L)]\)	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

\(S = \{-8, 2\}\)

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

\(L\)

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

\(H^2(Y, \mathbb{Z})\) with the Beauville-Bogomolov form, for \(Y\) deformation equivalent to the generalized Kummer variety \(K_{3}(T)\) of a complex torus \(T\) of dimension 2: \(U^3 \oplus \langle -8 \rangle\) (Rapagnetta).

\(U^{3} \oplus \langle -8 \rangle\)

Invariants

Gram tensor

Integral lattice

Indefinite lattice

Roots

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

Notes

Morphisms

References