K7 · Lattice Database

Rank	7
Signature \((n_+, n_-)\)	(7, 0)
Determinant	\(384\)

Rank

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

(7, 0)

\(384\)

Gram tensor

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

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

[[4, -2, 0, 0, 0, 0, 0], [-2, 4, -2, 2, 0, 0, 0], [0, -2, 4, 0, 0, 2, 0], [0, 2, 0, 4, 2, 2, 0], [0, 0, 0, 2, 4, 2, 2], [0, 0, 2, 2, 2, 4, 1], [0, 0, 0, 0, 2, 1, 4]]

Integral lattice

Parity	even
Discriminant group	\((\mathbb{Z}/2)^{4} \oplus \mathbb{Z}/24\)
Unimodular	no
Integral overlattices \(L \subseteq M \subseteq L^*\)	122
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)	\(\{2, 3\}\)
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 \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)	\(\zeta^\Sigma(X_n, s) = \zeta^\Sigma(s - 6)\, L^\Sigma(s - 3, \chi_{-384n})\)
Genus symbol	\(\mathrm{II}_{7,0}\ \bigl(2\colon 1^{2}:2^{-4}:[8^{1}]_{1};\ 3\colon 1^{6}\,3^{-1}\bigr)\)
Class number of the genus	4

Parity

even

Discriminant group

\((\mathbb{Z}/2)^{4} \oplus \mathbb{Z}/24\)

Unimodular

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

122

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

\(\{2, 3\}\)

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

\(\zeta^\Sigma(X_n, s) = \zeta^\Sigma(s - 6)\, L^\Sigma(s - 3, \chi_{-384n})\)

Genus symbol

\(\mathrm{II}_{7,0}\ \bigl(2\colon 1^{2}:2^{-4}:[8^{1}]_{1};\ 3\colon 1^{6}\,3^{-1}\bigr)\)

Class number of the genus

Minimum of \(b(x, x)\), \(x \neq 0\)	\(4\)
Kissing number	92
Order of the isometry group	\(7680 = 2^{9} \cdot 3 \cdot 5\)

Minimum of \(b(x, x)\), \(x \neq 0\)

\(4\)

Kissing number

Order of the isometry group

\(7680 = 2^{9} \cdot 3 \cdot 5\)

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)	no
Number of roots	42
\(R(L) = \mathbb{Z}\Phi(L)\)	\(\cong D_{5}(2) \oplus \langle 8 \rangle\)
Rank of \(R(L)\)	6
\([L : R(L)]\)	infinite: the rank of \(R(L)\) is less than the rank of \(L\)
\(R(L)\) is primitive in \(L\)	no: \(M / R(L) \cong \mathbb{Z}/2\) for \(M = L \cap (R(L) \otimes \mathbb{Q})\)

Root lattice: \(L = \mathbb{Z}\Phi(L)\)

Number of roots

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

\(\cong D_{5}(2) \oplus \langle 8 \rangle\)

Rank of \(R(L)\)

\([L : R(L)]\)

infinite: the rank of \(R(L)\) is less than the rank of \(L\)

\(R(L)\) is primitive in \(L\)

no: \(M / R(L) \cong \mathbb{Z}/2\) for \(M = L \cap (R(L) \otimes \mathbb{Q})\)

Component	Type	Values of \(b(r, r)\)	Number of roots	\(\mathbb{Z}\Phi_c\)
\(\Phi_{1}\)	\(D_{5}\)	\(\{4\}\)	40	\(\cong D_{5}(2)\)
\(\Phi_{2}\)	\(A_{1}\)	\(\{8\}\)	2	\(\cong \langle 8 \rangle\)

\(\alpha^{(c)}_j\) is simple root \(j\) of the component \(\Phi_c\).

Simple root	Coordinates	\(b(r, r)\)
\(\alpha^{(1)}_{1}\)	\(e_{4} - e_{5}\)	\(4\)
\(\alpha^{(1)}_{2}\)	\(e_{5}\)	\(4\)
\(\alpha^{(1)}_{3}\)	\(e_{2} - e_{4}\)	\(4\)
\(\alpha^{(1)}_{4}\)	\(e_{3}\)	\(4\)
\(\alpha^{(1)}_{5}\)	\(e_{1}\)	\(4\)
\(\alpha^{(2)}_{1}\)	\(e_{1} + 2e_{2} + e_{3} - 2e_{4} + 2e_{5} - 2e_{7}\)	\(8\)

The Gram tensor is the section GRAM of the entry KAPPA7 of the Catalogue of Lattices.

\(K_{7}\)

Invariants

Gram tensor

Integral lattice

Definite lattice

Theta series

Roots

Components of \(\Phi(L)\)

Simple roots

Notes

References