Tag 0258
\(D_{20}^{+}\)
D20+
Invariants
| Rank | 20 |
|---|---|
| Signature \((n_+, n_-)\) | (20, 0) |
| Determinant | \(1\) |
Gram tensor
The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{20}\).
| \(e_{1}\) | \(e_{2}\) | \(e_{3}\) | \(e_{4}\) | \(e_{5}\) | \(e_{6}\) | \(e_{7}\) | \(e_{8}\) | \(e_{9}\) | \(e_{10}\) | \(e_{11}\) | \(e_{12}\) | \(e_{13}\) | \(e_{14}\) | \(e_{15}\) | \(e_{16}\) | \(e_{17}\) | \(e_{18}\) | \(e_{19}\) | \(e_{20}\) | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(e_{1}\) | 2 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 |
| \(e_{2}\) | 1 | 2 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 |
| \(e_{3}\) | 1 | 1 | 2 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 |
| \(e_{4}\) | -1 | 0 | -1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 |
| \(e_{5}\) | 1 | 1 | 1 | 0 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | -1 |
| \(e_{6}\) | 1 | 1 | 1 | 0 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | -1 |
| \(e_{7}\) | 1 | 1 | 1 | 0 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | -1 |
| \(e_{8}\) | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | -1 |
| \(e_{9}\) | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | -1 |
| \(e_{10}\) | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | -1 |
| \(e_{11}\) | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | -1 |
| \(e_{12}\) | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | -1 |
| \(e_{13}\) | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 0 | -1 |
| \(e_{14}\) | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 0 | -1 |
| \(e_{15}\) | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 0 | -1 |
| \(e_{16}\) | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 0 | -1 |
| \(e_{17}\) | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 0 | -1 |
| \(e_{18}\) | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 5 | 3 | -3 |
| \(e_{19}\) | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 5 | -2 |
| \(e_{20}\) | -1 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -3 | -2 | 5 |
[[2, 1, 1, -1, 1, 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, 1, 1, 1, 1, 0, 0], [1, 1, 2, -1, 1, 1, 1, 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, 0, 0, 0, 0, -1, 0, 0], [1, 1, 1, 0, 2, 1, 1, 1, 1, 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, 1, 1, 1, 1, 0, -1], [1, 1, 1, 0, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, -1], [1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, -1], [1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, -1], [1, 1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 0, -1], [1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 0, -1], [1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 0, -1], [1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 0, -1], [1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 0, -1], [1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 0, -1], [1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 0, -1], [1, 1, 1, 0, 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, 1, 1, 1, 1, 1, 1, 1, 1, 5, 3, -3], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 5, -2], [-1, 0, 0, 0, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -3, -2, 5]]
Integral lattice
| Parity | odd |
|---|---|
| Discriminant group | \(0\) |
| Unimodular | yes |
| Integral overlattices \(L \subseteq M \subseteq L^*\) | 1 |
| Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\) | \(\{2\}\) |
| Character of the discriminant \(\chi_D\), \(D = (-1)^{10} \det L\) | trivial |
| Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\) | \(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 19)\, \zeta^\Sigma(s - 10)}{\zeta^\Sigma(s - 9)}\) |
| Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\) | \(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 19)}{\zeta^\Sigma(s - 9)}\) |
| Genus symbol | \(\mathrm{I}_{20,0}\) |
Definite lattice
| Minimum of \(b(x, x)\), \(x \neq 0\) | \(2\) |
|---|---|
| Kissing number | 760 |
Theta series
\[ \sum_{x \in L} q^{b(x, x)} = 1 + 760q^{2} + 77560q^{4} + O(q^{5}) \]Roots
| Root lattice: \(L = \mathbb{Z}\Phi(L)\) | no |
|---|---|
| Number of roots | 760 |
| \(R(L) = \mathbb{Z}\Phi(L)\) | \(\cong D_{20}\) |
| Rank of \(R(L)\) | 20 |
| \([L : R(L)]\) | 2 |
| \(R(L)\) is primitive in \(L\) | no: \(L / R(L) \cong \mathbb{Z}/2\) |
Components of \(\Phi(L)\)
| Component | Type | Values of \(b(r, r)\) | Number of roots | \(\mathbb{Z}\Phi_c\) |
|---|---|---|---|---|
| \(\Phi_{1}\) | \(D_{20}\) | \(\{2\}\) | 760 | \(\cong D_{20}\) |
Simple roots
| Simple root | Coordinates | \(b(r, r)\) |
|---|---|---|
| \(\alpha_{1}\) | \(e_{1} - e_{2} + e_{4} + e_{18} + e_{20}\) | \(2\) |
| \(\alpha_{2}\) | \(e_{1} - e_{18} + e_{19}\) | \(2\) |
| \(\alpha_{3}\) | \(5e_{1} + 6e_{3} + 6e_{4} - e_{5} - e_{6} - e_{7} - e_{8} - e_{9} - e_{10} - e_{11} - e_{12} - e_{13} - e_{14} - e_{15} - e_{16} - e_{17} + e_{18} - e_{20}\) | \(2\) |
| \(\alpha_{4}\) | \(e_{17}\) | \(2\) |
| \(\alpha_{5}\) | \(e_{16} - e_{17}\) | \(2\) |
| \(\alpha_{6}\) | \(e_{15} - e_{16}\) | \(2\) |
| \(\alpha_{7}\) | \(e_{14} - e_{15}\) | \(2\) |
| \(\alpha_{8}\) | \(e_{13} - e_{14}\) | \(2\) |
| \(\alpha_{9}\) | \(e_{12} - e_{13}\) | \(2\) |
| \(\alpha_{10}\) | \(e_{11} - e_{12}\) | \(2\) |
| \(\alpha_{11}\) | \(e_{10} - e_{11}\) | \(2\) |
| \(\alpha_{12}\) | \(e_{9} - e_{10}\) | \(2\) |
| \(\alpha_{13}\) | \(e_{8} - e_{9}\) | \(2\) |
| \(\alpha_{14}\) | \(e_{7} - e_{8}\) | \(2\) |
| \(\alpha_{15}\) | \(e_{6} - e_{7}\) | \(2\) |
| \(\alpha_{16}\) | \(e_{5} - e_{6}\) | \(2\) |
| \(\alpha_{17}\) | \(e_{3} - e_{5}\) | \(2\) |
| \(\alpha_{18}\) | \(e_{4}\) | \(2\) |
| \(\alpha_{19}\) | \(e_{1} - e_{2}\) | \(2\) |
| \(\alpha_{20}\) | \(e_{2} - e_{3} - e_{4}\) | \(2\) |
Notes
The basis is an LLL-reduced basis.