Tag 0223
\(A_{17}^{*}\)
A17*
Invariants
| Rank | 17 |
|---|---|
| Signature \((n_+, n_-)\) | (17, 0) |
| Determinant | \(\tfrac{1}{18}\) |
Gram tensor
The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{17}\).
| \(e_{1}\) | \(e_{2}\) | \(e_{3}\) | \(e_{4}\) | \(e_{5}\) | \(e_{6}\) | \(e_{7}\) | \(e_{8}\) | \(e_{9}\) | \(e_{10}\) | \(e_{11}\) | \(e_{12}\) | \(e_{13}\) | \(e_{14}\) | \(e_{15}\) | \(e_{16}\) | \(e_{17}\) | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(e_{1}\) | 17/18 | 8/9 | 5/6 | 7/9 | 13/18 | 2/3 | 11/18 | 5/9 | 1/2 | 4/9 | 7/18 | 1/3 | 5/18 | 2/9 | 1/6 | 1/9 | 1/18 |
| \(e_{2}\) | 8/9 | 16/9 | 5/3 | 14/9 | 13/9 | 4/3 | 11/9 | 10/9 | 1 | 8/9 | 7/9 | 2/3 | 5/9 | 4/9 | 1/3 | 2/9 | 1/9 |
| \(e_{3}\) | 5/6 | 5/3 | 5/2 | 7/3 | 13/6 | 2 | 11/6 | 5/3 | 3/2 | 4/3 | 7/6 | 1 | 5/6 | 2/3 | 1/2 | 1/3 | 1/6 |
| \(e_{4}\) | 7/9 | 14/9 | 7/3 | 28/9 | 26/9 | 8/3 | 22/9 | 20/9 | 2 | 16/9 | 14/9 | 4/3 | 10/9 | 8/9 | 2/3 | 4/9 | 2/9 |
| \(e_{5}\) | 13/18 | 13/9 | 13/6 | 26/9 | 65/18 | 10/3 | 55/18 | 25/9 | 5/2 | 20/9 | 35/18 | 5/3 | 25/18 | 10/9 | 5/6 | 5/9 | 5/18 |
| \(e_{6}\) | 2/3 | 4/3 | 2 | 8/3 | 10/3 | 4 | 11/3 | 10/3 | 3 | 8/3 | 7/3 | 2 | 5/3 | 4/3 | 1 | 2/3 | 1/3 |
| \(e_{7}\) | 11/18 | 11/9 | 11/6 | 22/9 | 55/18 | 11/3 | 77/18 | 35/9 | 7/2 | 28/9 | 49/18 | 7/3 | 35/18 | 14/9 | 7/6 | 7/9 | 7/18 |
| \(e_{8}\) | 5/9 | 10/9 | 5/3 | 20/9 | 25/9 | 10/3 | 35/9 | 40/9 | 4 | 32/9 | 28/9 | 8/3 | 20/9 | 16/9 | 4/3 | 8/9 | 4/9 |
| \(e_{9}\) | 1/2 | 1 | 3/2 | 2 | 5/2 | 3 | 7/2 | 4 | 9/2 | 4 | 7/2 | 3 | 5/2 | 2 | 3/2 | 1 | 1/2 |
| \(e_{10}\) | 4/9 | 8/9 | 4/3 | 16/9 | 20/9 | 8/3 | 28/9 | 32/9 | 4 | 40/9 | 35/9 | 10/3 | 25/9 | 20/9 | 5/3 | 10/9 | 5/9 |
| \(e_{11}\) | 7/18 | 7/9 | 7/6 | 14/9 | 35/18 | 7/3 | 49/18 | 28/9 | 7/2 | 35/9 | 77/18 | 11/3 | 55/18 | 22/9 | 11/6 | 11/9 | 11/18 |
| \(e_{12}\) | 1/3 | 2/3 | 1 | 4/3 | 5/3 | 2 | 7/3 | 8/3 | 3 | 10/3 | 11/3 | 4 | 10/3 | 8/3 | 2 | 4/3 | 2/3 |
| \(e_{13}\) | 5/18 | 5/9 | 5/6 | 10/9 | 25/18 | 5/3 | 35/18 | 20/9 | 5/2 | 25/9 | 55/18 | 10/3 | 65/18 | 26/9 | 13/6 | 13/9 | 13/18 |
| \(e_{14}\) | 2/9 | 4/9 | 2/3 | 8/9 | 10/9 | 4/3 | 14/9 | 16/9 | 2 | 20/9 | 22/9 | 8/3 | 26/9 | 28/9 | 7/3 | 14/9 | 7/9 |
| \(e_{15}\) | 1/6 | 1/3 | 1/2 | 2/3 | 5/6 | 1 | 7/6 | 4/3 | 3/2 | 5/3 | 11/6 | 2 | 13/6 | 7/3 | 5/2 | 5/3 | 5/6 |
| \(e_{16}\) | 1/9 | 2/9 | 1/3 | 4/9 | 5/9 | 2/3 | 7/9 | 8/9 | 1 | 10/9 | 11/9 | 4/3 | 13/9 | 14/9 | 5/3 | 16/9 | 8/9 |
| \(e_{17}\) | 1/18 | 1/9 | 1/6 | 2/9 | 5/18 | 1/3 | 7/18 | 4/9 | 1/2 | 5/9 | 11/18 | 2/3 | 13/18 | 7/9 | 5/6 | 8/9 | 17/18 |
[[17/18, 8/9, 5/6, 7/9, 13/18, 2/3, 11/18, 5/9, 1/2, 4/9, 7/18, 1/3, 5/18, 2/9, 1/6, 1/9, 1/18], [8/9, 16/9, 5/3, 14/9, 13/9, 4/3, 11/9, 10/9, 1, 8/9, 7/9, 2/3, 5/9, 4/9, 1/3, 2/9, 1/9], [5/6, 5/3, 5/2, 7/3, 13/6, 2, 11/6, 5/3, 3/2, 4/3, 7/6, 1, 5/6, 2/3, 1/2, 1/3, 1/6], [7/9, 14/9, 7/3, 28/9, 26/9, 8/3, 22/9, 20/9, 2, 16/9, 14/9, 4/3, 10/9, 8/9, 2/3, 4/9, 2/9], [13/18, 13/9, 13/6, 26/9, 65/18, 10/3, 55/18, 25/9, 5/2, 20/9, 35/18, 5/3, 25/18, 10/9, 5/6, 5/9, 5/18], [2/3, 4/3, 2, 8/3, 10/3, 4, 11/3, 10/3, 3, 8/3, 7/3, 2, 5/3, 4/3, 1, 2/3, 1/3], [11/18, 11/9, 11/6, 22/9, 55/18, 11/3, 77/18, 35/9, 7/2, 28/9, 49/18, 7/3, 35/18, 14/9, 7/6, 7/9, 7/18], [5/9, 10/9, 5/3, 20/9, 25/9, 10/3, 35/9, 40/9, 4, 32/9, 28/9, 8/3, 20/9, 16/9, 4/3, 8/9, 4/9], [1/2, 1, 3/2, 2, 5/2, 3, 7/2, 4, 9/2, 4, 7/2, 3, 5/2, 2, 3/2, 1, 1/2], [4/9, 8/9, 4/3, 16/9, 20/9, 8/3, 28/9, 32/9, 4, 40/9, 35/9, 10/3, 25/9, 20/9, 5/3, 10/9, 5/9], [7/18, 7/9, 7/6, 14/9, 35/18, 7/3, 49/18, 28/9, 7/2, 35/9, 77/18, 11/3, 55/18, 22/9, 11/6, 11/9, 11/18], [1/3, 2/3, 1, 4/3, 5/3, 2, 7/3, 8/3, 3, 10/3, 11/3, 4, 10/3, 8/3, 2, 4/3, 2/3], [5/18, 5/9, 5/6, 10/9, 25/18, 5/3, 35/18, 20/9, 5/2, 25/9, 55/18, 10/3, 65/18, 26/9, 13/6, 13/9, 13/18], [2/9, 4/9, 2/3, 8/9, 10/9, 4/3, 14/9, 16/9, 2, 20/9, 22/9, 8/3, 26/9, 28/9, 7/3, 14/9, 7/9], [1/6, 1/3, 1/2, 2/3, 5/6, 1, 7/6, 4/3, 3/2, 5/3, 11/6, 2, 13/6, 7/3, 5/2, 5/3, 5/6], [1/9, 2/9, 1/3, 4/9, 5/9, 2/3, 7/9, 8/9, 1, 10/9, 11/9, 4/3, 13/9, 14/9, 5/3, 16/9, 8/9], [1/18, 1/9, 1/6, 2/9, 5/18, 1/3, 7/18, 4/9, 1/2, 5/9, 11/18, 2/3, 13/18, 7/9, 5/6, 8/9, 17/18]]
Definite lattice
| Minimum of \(b(x, x)\), \(x \neq 0\) | \(\tfrac{17}{18}\) |
|---|---|
| Kissing number | 36 |
| Order of the isometry group | \(12804747411456000 = 2^{17} \cdot 3^{8} \cdot 5^{3} \cdot 7^{2} \cdot 11 \cdot 13 \cdot 17\) |
Roots
| Root lattice: \(L = \mathbb{Z}\Phi(L)\) | no |
|---|---|
| Number of roots | 306 |
| \(R(L) = \mathbb{Z}\Phi(L)\) | \(\cong A_{17}\) |
| Rank of \(R(L)\) | 17 |
| \([L : R(L)]\) | 18 |
| \(R(L)\) is primitive in \(L\) | no: \(L / R(L) \cong \mathbb{Z}/18\) |
Components of \(\Phi(L)\)
| Component | Type | Values of \(b(r, r)\) | Number of roots | \(\mathbb{Z}\Phi_c\) |
|---|---|---|---|---|
| \(\Phi_{1}\) | \(A_{17}\) | \(\{2\}\) | 306 | \(\cong A_{17}\) |
Simple roots
| Simple root | Coordinates | \(b(r, r)\) |
|---|---|---|
| \(\alpha_{1}\) | \(e_{1} + e_{17}\) | \(2\) |
| \(\alpha_{2}\) | \(e_{16} - 2e_{17}\) | \(2\) |
| \(\alpha_{3}\) | \(e_{15} - 2e_{16} + e_{17}\) | \(2\) |
| \(\alpha_{4}\) | \(e_{14} - 2e_{15} + e_{16}\) | \(2\) |
| \(\alpha_{5}\) | \(e_{13} - 2e_{14} + e_{15}\) | \(2\) |
| \(\alpha_{6}\) | \(e_{12} - 2e_{13} + e_{14}\) | \(2\) |
| \(\alpha_{7}\) | \(e_{11} - 2e_{12} + e_{13}\) | \(2\) |
| \(\alpha_{8}\) | \(e_{10} - 2e_{11} + e_{12}\) | \(2\) |
| \(\alpha_{9}\) | \(e_{9} - 2e_{10} + e_{11}\) | \(2\) |
| \(\alpha_{10}\) | \(e_{8} - 2e_{9} + e_{10}\) | \(2\) |
| \(\alpha_{11}\) | \(e_{7} - 2e_{8} + e_{9}\) | \(2\) |
| \(\alpha_{12}\) | \(e_{6} - 2e_{7} + e_{8}\) | \(2\) |
| \(\alpha_{13}\) | \(e_{5} - 2e_{6} + e_{7}\) | \(2\) |
| \(\alpha_{14}\) | \(e_{4} - 2e_{5} + e_{6}\) | \(2\) |
| \(\alpha_{15}\) | \(e_{3} - 2e_{4} + e_{5}\) | \(2\) |
| \(\alpha_{16}\) | \(e_{2} - 2e_{3} + e_{4}\) | \(2\) |
| \(\alpha_{17}\) | \(e_{1} - 2e_{2} + e_{3}\) | \(2\) |