K9 · Lattice Database

Rank	9
Signature \((n_+, n_-)\)	(9, 0)
Determinant	\(864\)

Rank

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

(9, 0)

\(864\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{9}\).

	\(e_{1}\)	\(e_{2}\)	\(e_{3}\)	\(e_{4}\)	\(e_{5}\)	\(e_{6}\)	\(e_{7}\)	\(e_{8}\)	\(e_{9}\)
\(e_{1}\)	4	-2	0	0	0	0	0	0	0
\(e_{2}\)	-2	4	-2	2	0	0	0	0	0
\(e_{3}\)	0	-2	4	0	0	2	0	2	2
\(e_{4}\)	0	2	0	4	2	2	0	2	1
\(e_{5}\)	0	0	0	2	4	2	2	1	1
\(e_{6}\)	0	0	2	2	2	4	1	2	2
\(e_{7}\)	0	0	0	0	2	1	4	1	2
\(e_{8}\)	0	0	2	2	1	2	1	4	2
\(e_{9}\)	0	0	2	1	1	2	2	2	4

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

Integral lattice

Parity	even
Discriminant group	\((\mathbb{Z}/6)^{2} \oplus \mathbb{Z}/24\)
Unimodular	no
Integral overlattices \(L \subseteq M \subseteq L^*\)	55
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)	\(\{2, 3\}\)
Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)	\(\zeta^\Sigma(X_0, s) = \zeta^\Sigma(s - 8)\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)	\(\zeta^\Sigma(X_n, s) = \zeta^\Sigma(s - 8)\, L^\Sigma(s - 4, \chi_{864n})\)
Genus symbol	\(\mathrm{II}_{9,0}\ \bigl(2\colon 1^{6}:2^{-2}:[8^{1}]_{7};\ 3\colon 1^{6}\,3^{-3}\bigr)\)
Class number of the genus	34

Parity

even

Discriminant group

\((\mathbb{Z}/6)^{2} \oplus \mathbb{Z}/24\)

Unimodular

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

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

\(\{2, 3\}\)

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

\(\zeta^\Sigma(X_0, s) = \zeta^\Sigma(s - 8)\)

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

\(\zeta^\Sigma(X_n, s) = \zeta^\Sigma(s - 8)\, L^\Sigma(s - 4, \chi_{864n})\)

Genus symbol

\(\mathrm{II}_{9,0}\ \bigl(2\colon 1^{6}:2^{-2}:[8^{1}]_{7};\ 3\colon 1^{6}\,3^{-3}\bigr)\)

Class number of the genus

Minimum of \(b(x, x)\), \(x \neq 0\)	\(4\)
Kissing number	180
Order of the isometry group	\(2304 = 2^{8} \cdot 3^{2}\)

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

\(4\)

Kissing number

180

Order of the isometry group

\(2304 = 2^{8} \cdot 3^{2}\)

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)	no
Number of roots	12
\(R(L) = \mathbb{Z}\Phi(L)\)	\(\cong A_{3}(2)\)
Rank of \(R(L)\)	3
\([L : R(L)]\)	infinite: the rank of \(R(L)\) is less than the rank of \(L\)
\(R(L)\) is primitive in \(L\)	yes

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

Number of roots

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

\(\cong A_{3}(2)\)

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 \(L\)

yes

Component	Type	Values of \(b(r, r)\)	Number of roots	\(\mathbb{Z}\Phi_c\)
\(\Phi_{1}\)	\(A_{3}\)	\(\{4\}\)	12	\(\cong A_{3}(2)\)

Simple root	Coordinates	\(b(r, r)\)
\(\alpha_{1}\)	\(e_{3}\)	\(4\)
\(\alpha_{2}\)	\(e_{2}\)	\(4\)
\(\alpha_{3}\)	\(e_{1}\)	\(4\)

The Gram tensor is the section GRAM of the entry KAPPA9 of the Catalogue of Lattices.

\(K_{9}\)

Invariants

Gram tensor

Integral lattice

Definite lattice

Theta series

Roots

Components of \(\Phi(L)\)

Simple roots

Notes

References