Tag 0206

\(D_{16}^{+}\)

D16+

Invariants

Rank16
Signature \((n_+, n_-)\)(16, 0)
Determinant\(1\)

Gram tensor

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

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)\(e_{13}\)\(e_{14}\)\(e_{15}\)\(e_{16}\)
\(e_{1}\)211-11111111111-1-1
\(e_{2}\)1210111111111100
\(e_{3}\)112-1111111111100
\(e_{4}\)-10-12000000000-100
\(e_{5}\)111021111111110-1
\(e_{6}\)111012111111110-1
\(e_{7}\)111011211111110-1
\(e_{8}\)111011121111110-1
\(e_{9}\)111011112111110-1
\(e_{10}\)111011111211110-1
\(e_{11}\)111011111121110-1
\(e_{12}\)111011111112110-1
\(e_{13}\)111011111111210-1
\(e_{14}\)111-111111111142-2
\(e_{15}\)-100000000000024-1
\(e_{16}\)-1000-1-1-1-1-1-1-1-1-1-2-14
 
[[2, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1], [1, 2, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0], [1, 1, 2, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0], [-1, 0, -1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0], [1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, -1], [1, 1, 1, 0, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 0, -1], [1, 1, 1, 0, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 0, -1], [1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 0, -1], [1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 0, -1], [1, 1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 0, -1], [1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 0, -1], [1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 0, -1], [1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, -1], [1, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 2, -2], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, -1], [-1, 0, 0, 0, -1, -1, -1, -1, -1, -1, -1, -1, -1, -2, -1, 4]]

Integral lattice

Parityeven
Discriminant group\(0\)
Unimodularyes
Integral overlattices \(L \subseteq M \subseteq L^*\)1
Nikulin invariants \((r, a, \delta)\)\((16, 0, 0)\)
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\(\mathrm{II}_{16,0}\)

Definite lattice

Minimum of \(b(x, x)\), \(x \neq 0\)\(2\)
Kissing number480
Order of the isometry group\(685597979049984000 = 2^{30} \cdot 3^{6} \cdot 5^{3} \cdot 7^{2} \cdot 11 \cdot 13\)

Theta series

\[ \sum_{x \in L} q^{b(x, x)} = 1 + 480q^{2} + 61920q^{4} + O(q^{5}) \]

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)no
Number of roots480
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong D_{16}\)
Rank of \(R(L)\)16
\([L : R(L)]\)2
\(R(L)\) is primitive in \(L\)no: \(L / R(L) \cong \mathbb{Z}/2\)

Components of \(\Phi(L)\)

ComponentTypeValues of \(b(r, r)\)Number of roots\(\mathbb{Z}\Phi_c\)
\(\Phi_{1}\)\(D_{16}\)\(\{2\}\)480\(\cong D_{16}\)

Simple roots

Simple rootCoordinates\(b(r, r)\)
\(\alpha_{1}\)\(e_{1} - e_{2} + e_{4} + e_{14} + e_{16}\)\(2\)
\(\alpha_{2}\)\(e_{1} - e_{14} + e_{15}\)\(2\)
\(\alpha_{3}\)\(3e_{1} + 4e_{3} + 4e_{4} - e_{5} - e_{6} - e_{7} - e_{8} - e_{9} - e_{10} - e_{11} - e_{12} - e_{13} + e_{14} - e_{16}\)\(2\)
\(\alpha_{4}\)\(e_{13}\)\(2\)
\(\alpha_{5}\)\(e_{12} - e_{13}\)\(2\)
\(\alpha_{6}\)\(e_{11} - e_{12}\)\(2\)
\(\alpha_{7}\)\(e_{10} - e_{11}\)\(2\)
\(\alpha_{8}\)\(e_{9} - e_{10}\)\(2\)
\(\alpha_{9}\)\(e_{8} - e_{9}\)\(2\)
\(\alpha_{10}\)\(e_{7} - e_{8}\)\(2\)
\(\alpha_{11}\)\(e_{6} - e_{7}\)\(2\)
\(\alpha_{12}\)\(e_{5} - e_{6}\)\(2\)
\(\alpha_{13}\)\(e_{3} - e_{5}\)\(2\)
\(\alpha_{14}\)\(e_{4}\)\(2\)
\(\alpha_{15}\)\(e_{1} - e_{2}\)\(2\)
\(\alpha_{16}\)\(e_{2} - e_{3} - e_{4}\)\(2\)

Notes

The basis is an LLL-reduced basis.