Tag 0244

\(D_{19}\)

D19

Invariants

Rank19
Signature \((n_+, n_-)\)(19, 0)
Determinant\(4\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{19}\).

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)\(e_{13}\)\(e_{14}\)\(e_{15}\)\(e_{16}\)\(e_{17}\)\(e_{18}\)\(e_{19}\)
\(e_{1}\)2-100000000000000000
\(e_{2}\)-12-10000000000000000
\(e_{3}\)0-12-1000000000000000
\(e_{4}\)00-12-100000000000000
\(e_{5}\)000-12-10000000000000
\(e_{6}\)0000-12-1000000000000
\(e_{7}\)00000-12-100000000000
\(e_{8}\)000000-12-10000000000
\(e_{9}\)0000000-12-1000000000
\(e_{10}\)00000000-12-100000000
\(e_{11}\)000000000-12-10000000
\(e_{12}\)0000000000-12-1000000
\(e_{13}\)00000000000-12-100000
\(e_{14}\)000000000000-12-10000
\(e_{15}\)0000000000000-12-1000
\(e_{16}\)00000000000000-12-100
\(e_{17}\)000000000000000-12-1-1
\(e_{18}\)0000000000000000-120
\(e_{19}\)0000000000000000-102
 
[[2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 2, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 2, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 2, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 2, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 2, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 2, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 2, -1, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 2, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 2]]

Integral lattice

Parityeven
Discriminant group\(\mathbb{Z}/4\)
Unimodularno
Integral overlattices \(L \subseteq M \subseteq L^*\)2
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)\(\{2\}\)
Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)\(\zeta^\Sigma(X_0, s) = \zeta^\Sigma(s - 18)\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)\(\zeta^\Sigma(X_n, s) = \zeta^\Sigma(s - 18)\, L^\Sigma(s - 9, \chi_{-4n})\)
Genus symbol\(\mathrm{II}_{19,0}\ \bigl(2\colon 1^{-18}:[4^{-1}]_{3}\bigr)\)

Definite lattice

Minimum of \(b(x, x)\), \(x \neq 0\)\(2\)
Kissing number684
Order of the isometry group\(63777066403145711616000 = 2^{35} \cdot 3^{8} \cdot 5^{3} \cdot 7^{2} \cdot 11 \cdot 13 \cdot 17 \cdot 19\)

Theta series

\[ \sum_{x \in L} q^{b(x, x)} = 1 + 684q^{2} + 62054q^{4} + O(q^{5}) \]

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)yes
\(L = \mathbb{Z}\Phi_S(L)\) for\(S = \{2\}\)
Number of roots722
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong D_{19}\)
Rank of \(R(L)\)19
\([L : R(L)]\)1
\(R(L)\) is primitive in \(L\)yes

Components of \(\Phi(L)\)

ComponentTypeValues of \(b(r, r)\)Number of roots\(\mathbb{Z}\Phi_c\)
\(\Phi_{1}\)\(C_{19}\)\(\{2, 4\}\)722\(\cong D_{19}\)

Simple roots

Simple rootCoordinates\(b(r, r)\)
\(\alpha_{1}\)\(e_{1}\)\(2\)
\(\alpha_{2}\)\(e_{2}\)\(2\)
\(\alpha_{3}\)\(e_{3}\)\(2\)
\(\alpha_{4}\)\(e_{4}\)\(2\)
\(\alpha_{5}\)\(e_{5}\)\(2\)
\(\alpha_{6}\)\(e_{6}\)\(2\)
\(\alpha_{7}\)\(e_{7}\)\(2\)
\(\alpha_{8}\)\(e_{8}\)\(2\)
\(\alpha_{9}\)\(e_{9}\)\(2\)
\(\alpha_{10}\)\(e_{10}\)\(2\)
\(\alpha_{11}\)\(e_{11}\)\(2\)
\(\alpha_{12}\)\(e_{12}\)\(2\)
\(\alpha_{13}\)\(e_{13}\)\(2\)
\(\alpha_{14}\)\(e_{14}\)\(2\)
\(\alpha_{15}\)\(e_{15}\)\(2\)
\(\alpha_{16}\)\(e_{16}\)\(2\)
\(\alpha_{17}\)\(e_{17}\)\(2\)
\(\alpha_{18}\)\(e_{19}\)\(2\)
\(\alpha_{19}\)\(e_{18} - e_{19}\)\(4\)
  • 0250 \(D_{19}^{*}\) The dual lattice \(L^*\).