Lambda16 · Lattice Database

Rank	16
Signature \((n_+, n_-)\)	(16, 0)
Determinant	\(256\)

Rank

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

(16, 0)

\(256\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{16}\).

	\(e_{1}\)	\(e_{2}\)	\(e_{3}\)	\(e_{4}\)	\(e_{5}\)	\(e_{6}\)	\(e_{7}\)	\(e_{8}\)	\(e_{9}\)	\(e_{10}\)	\(e_{11}\)	\(e_{12}\)	\(e_{13}\)	\(e_{14}\)	\(e_{15}\)	\(e_{16}\)
\(e_{1}\)	4	-2	0	0	0	0	0	0	0	0	0	0	0	1	1	0
\(e_{2}\)	-2	4	-2	2	0	0	0	0	0	0	0	0	-1	0	0	1
\(e_{3}\)	0	-2	4	0	0	2	0	0	0	0	0	0	2	1	1	0
\(e_{4}\)	0	2	0	4	2	2	0	0	0	0	0	0	0	1	1	2
\(e_{5}\)	0	0	0	2	4	2	0	0	2	1	0	0	0	0	0	2
\(e_{6}\)	0	0	2	2	2	4	2	2	1	2	0	0	1	1	2	2
\(e_{7}\)	0	0	0	0	0	2	4	2	0	2	0	0	0	-1	1	0
\(e_{8}\)	0	0	0	0	0	2	2	4	0	2	0	0	1	0	2	1
\(e_{9}\)	0	0	0	0	2	1	0	0	4	2	0	0	0	0	0	2
\(e_{10}\)	0	0	0	0	1	2	2	2	2	4	2	2	1	1	2	2
\(e_{11}\)	0	0	0	0	0	0	0	0	0	2	4	2	2	1	1	0
\(e_{12}\)	0	0	0	0	0	0	0	0	0	2	2	4	1	2	2	1
\(e_{13}\)	0	-1	2	0	0	1	0	1	0	1	2	1	4	0	2	0
\(e_{14}\)	1	0	1	1	0	1	-1	0	0	1	1	2	0	4	2	2
\(e_{15}\)	1	0	1	1	0	2	1	2	0	2	1	2	2	2	4	2
\(e_{16}\)	0	1	0	2	2	2	0	1	2	2	0	1	0	2	2	4

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

Integral lattice

Parity	even
Discriminant group	\((\mathbb{Z}/2)^{8}\)
Unimodular	no
Integral overlattices \(L \subseteq M \subseteq L^*\)	not decided
Nikulin invariants \((r, a, \delta)\)	\((16, 8, 0)\)
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)	\(\{2\}\)
Character of the discriminant \(\chi_D\), \(D = (-1)^{8} \det L\)	trivial
Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)	\(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 15)\, \zeta^\Sigma(s - 8)}{\zeta^\Sigma(s - 7)}\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)	\(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 15)}{\zeta^\Sigma(s - 7)}\)
Genus symbol	\(\mathrm{II}_{16,0}\ \bigl(2\colon 1^{8}:2^{8}\bigr)\)

Parity

even

Discriminant group

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

Unimodular

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

not decided

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

\((16, 8, 0)\)

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

\(\{2\}\)

Character of the discriminant \(\chi_D\), \(D = (-1)^{8} \det L\)

trivial

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

\(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 15)\, \zeta^\Sigma(s - 8)}{\zeta^\Sigma(s - 7)}\)

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

\(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 15)}{\zeta^\Sigma(s - 7)}\)

Genus symbol

\(\mathrm{II}_{16,0}\ \bigl(2\colon 1^{8}:2^{8}\bigr)\)

Minimum of \(b(x, x)\), \(x \neq 0\)	\(4\)
Kissing number	4320
Order of the isometry group	\(89181388800 = 2^{21} \cdot 3^{5} \cdot 5^{2} \cdot 7\)

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

\(4\)

Kissing number

4320

Order of the isometry group

\(89181388800 = 2^{21} \cdot 3^{5} \cdot 5^{2} \cdot 7\)

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 LAMBDA16 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

\(\Lambda_{16}\)

Invariants

Gram tensor

Integral lattice

Definite lattice

Theta series

Roots

Notes

References