U(2) + D4(-1)^3 · Lattice Database

Rank	14
Signature \((n_+, n_-)\)	(1, 13)
Determinant	\(-256\)

Rank

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

(1, 13)

\(-256\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{14}\).

In this basis, \(L = {}\)\(\mathbb{Z}\{e_{1}, e_{2}\}\) \(\oplus\) \(\mathbb{Z}\{e_{3}, \dots, e_{6}\}\) \(\oplus\) \(\mathbb{Z}\{e_{7}, \dots, e_{10}\}\) \(\oplus\) \(\mathbb{Z}\{e_{11}, \dots, e_{14}\}\), an orthogonal sum.

	\(e_{1}\)	\(e_{2}\)	\(e_{3}\)	\(e_{4}\)	\(e_{5}\)	\(e_{6}\)	\(e_{7}\)	\(e_{8}\)	\(e_{9}\)	\(e_{10}\)	\(e_{11}\)	\(e_{12}\)	\(e_{13}\)	\(e_{14}\)
\(e_{1}\)	0	2	0	0	0	0	0	0	0	0	0	0	0	0
\(e_{2}\)	2	0	0	0	0	0	0	0	0	0	0	0	0	0
\(e_{3}\)	0	0	-2	1	0	0	0	0	0	0	0	0	0	0
\(e_{4}\)	0	0	1	-2	1	1	0	0	0	0	0	0	0	0
\(e_{5}\)	0	0	0	1	-2	0	0	0	0	0	0	0	0	0
\(e_{6}\)	0	0	0	1	0	-2	0	0	0	0	0	0	0	0
\(e_{7}\)	0	0	0	0	0	0	-2	1	0	0	0	0	0	0
\(e_{8}\)	0	0	0	0	0	0	1	-2	1	1	0	0	0	0
\(e_{9}\)	0	0	0	0	0	0	0	1	-2	0	0	0	0	0
\(e_{10}\)	0	0	0	0	0	0	0	1	0	-2	0	0	0	0
\(e_{11}\)	0	0	0	0	0	0	0	0	0	0	-2	1	0	0
\(e_{12}\)	0	0	0	0	0	0	0	0	0	0	1	-2	1	1
\(e_{13}\)	0	0	0	0	0	0	0	0	0	0	0	1	-2	0
\(e_{14}\)	0	0	0	0	0	0	0	0	0	0	0	1	0	-2

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

Integral lattice

Parity	even
Discriminant group	\((\mathbb{Z}/2)^{8}\)
Unimodular	no
Integral overlattices \(L \subseteq M \subseteq L^*\)	not decided
Nikulin invariants \((r, a, \delta)\)	\((14, 8, 0)\)
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)	\(\{2\}\)
Character of the discriminant \(\chi_D\), \(D = (-1)^{7} \det L\)	trivial
Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)	\(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 13)\, \zeta^\Sigma(s - 7)}{\zeta^\Sigma(s - 6)}\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)	\(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 13)}{\zeta^\Sigma(s - 6)}\)
Genus symbol	\(\mathrm{II}_{1,13}\ \bigl(2\colon 1^{-6}:2^{-8}\bigr)\)

Parity

even

Discriminant group

\((\mathbb{Z}/2)^{8}\)

Unimodular

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

not decided

Nikulin invariants \((r, a, \delta)\)

\((14, 8, 0)\)

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

\(\{2\}\)

Character of the discriminant \(\chi_D\), \(D = (-1)^{7} \det L\)

trivial

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

\(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 13)\, \zeta^\Sigma(s - 7)}{\zeta^\Sigma(s - 6)}\)

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

\(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 13)}{\zeta^\Sigma(s - 6)}\)

Genus symbol

\(\mathrm{II}_{1,13}\ \bigl(2\colon 1^{-6}:2^{-8}\bigr)\)

Isotropic	yes

Reflective	yes declared

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)	yes
\(L = \mathbb{Z}\Phi_S(L)\) for	\(S = \{-2, 4\}\)
\(R(L) = \mathbb{Z}\Phi(L)\)	\(L\)
Rank of \(R(L)\)	14
\([L : R(L)]\)	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

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

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

\(L\)

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	\(b(r, r)\)
\(r_{1}\)	\(e_{3}\)	\(-2\)
\(r_{2}\)	\(e_{4}\)	\(-2\)
\(r_{3}\)	\(e_{5}\)	\(-2\)
\(r_{4}\)	\(e_{6}\)	\(-2\)
\(r_{5}\)	\(e_{7}\)	\(-2\)
\(r_{6}\)	\(e_{8}\)	\(-2\)
\(r_{7}\)	\(e_{9}\)	\(-2\)
\(r_{8}\)	\(e_{10}\)	\(-2\)
\(r_{9}\)	\(e_{11}\)	\(-2\)
\(r_{10}\)	\(e_{12}\)	\(-2\)
\(r_{11}\)	\(e_{13}\)	\(-2\)
\(r_{12}\)	\(e_{14}\)	\(-2\)
\(r_{13}\)	\(e_{1} + e_{2}\)	\(4\)
\(r_{14}\)	\(e_{1} + e_{3}\)	\(-2\)

This is the even hyperbolic 2-elementary lattice with invariants \((r, a, \delta) = (14, 8, 0)\).

Its invariants satisfy \(r + a = 22\), and it is reflective (Alexeev, Theorem 1.1).

\(U(2) \oplus D_{4}(-1)^{3}\)

Invariants

Gram tensor

Integral lattice

Indefinite lattice

Hyperbolic lattice

Roots

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

Notes

Morphisms

References