Lambda14 · Lattice Database

Rank	14
Signature \((n_+, n_-)\)	(14, 0)
Determinant	\(768\)

Rank

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

(14, 0)

\(768\)

Gram tensor

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

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

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

Integral lattice

Parity	even
Discriminant group	\((\mathbb{Z}/2)^{4} \oplus \mathbb{Z}/4 \oplus \mathbb{Z}/12\)
Unimodular	no
Integral overlattices \(L \subseteq M \subseteq L^*\)	605
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)	\(\{2, 3\}\)
Character of the discriminant \(\chi_D\), \(D = (-1)^{7} \det L\)	\(\chi_{-3} = \left(\frac{-3}{\cdot}\right)\)
Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)	\(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 13)\, L^\Sigma(s - 7, \chi_{-3})}{L^\Sigma(s - 6, \chi_{-3})}\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)	\(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 13)}{L^\Sigma(s - 6, \chi_{-3})}\)
Genus symbol	\(\mathrm{II}_{14,0}\ \bigl(2\colon 1^{8}:2^{4}:4^{-2};\ 3\colon 1^{13}\,3^{1}\bigr)\)

Parity

even

Discriminant group

\((\mathbb{Z}/2)^{4} \oplus \mathbb{Z}/4 \oplus \mathbb{Z}/12\)

Unimodular

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

605

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

\(\{2, 3\}\)

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

\(\chi_{-3} = \left(\frac{-3}{\cdot}\right)\)

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

\(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 13)\, L^\Sigma(s - 7, \chi_{-3})}{L^\Sigma(s - 6, \chi_{-3})}\)

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

\(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 13)}{L^\Sigma(s - 6, \chi_{-3})}\)

Genus symbol

\(\mathrm{II}_{14,0}\ \bigl(2\colon 1^{8}:2^{4}:4^{-2};\ 3\colon 1^{13}\,3^{1}\bigr)\)

Minimum of \(b(x, x)\), \(x \neq 0\)	\(4\)
Kissing number	1422
Order of the isometry group	\(884736 = 2^{15} \cdot 3^{3}\)

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

\(4\)

Kissing number

1422

Order of the isometry group

\(884736 = 2^{15} \cdot 3^{3}\)

Root lattice: \(L = \mathbb{Z}\Phi(L)\)	no
Number of roots	0
\(R(L) = \mathbb{Z}\Phi(L)\)	\(0\)
Rank of \(R(L)\)	0
\([L : R(L)]\)	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

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

\(0\)

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

The Gram tensor is the section GRAM of the entry LAMBDA14 of the Catalogue of Lattices.

The catalogue entry gives this reference text: 1. SPLAG p. 157. 2. W.Plesken, M.Pohst, Constructing integral lattices with prescribed minimum II, Math. Comp. Vol 60, N. 202 (1993) 817-825

\(\Lambda_{14}\)

Invariants

Gram tensor

Integral lattice

Definite lattice

Theta series

Roots

Notes

References