Tag 0262
\(A_{20}^{*}\)
A20*
Invariants
| Rank | 20 |
|---|---|
| Signature \((n_+, n_-)\) | (20, 0) |
| Determinant | \(\tfrac{1}{21}\) |
Gram tensor
The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{20}\).
| \(e_{1}\) | \(e_{2}\) | \(e_{3}\) | \(e_{4}\) | \(e_{5}\) | \(e_{6}\) | \(e_{7}\) | \(e_{8}\) | \(e_{9}\) | \(e_{10}\) | \(e_{11}\) | \(e_{12}\) | \(e_{13}\) | \(e_{14}\) | \(e_{15}\) | \(e_{16}\) | \(e_{17}\) | \(e_{18}\) | \(e_{19}\) | \(e_{20}\) | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(e_{1}\) | 20/21 | 19/21 | 6/7 | 17/21 | 16/21 | 5/7 | 2/3 | 13/21 | 4/7 | 11/21 | 10/21 | 3/7 | 8/21 | 1/3 | 2/7 | 5/21 | 4/21 | 1/7 | 2/21 | 1/21 |
| \(e_{2}\) | 19/21 | 38/21 | 12/7 | 34/21 | 32/21 | 10/7 | 4/3 | 26/21 | 8/7 | 22/21 | 20/21 | 6/7 | 16/21 | 2/3 | 4/7 | 10/21 | 8/21 | 2/7 | 4/21 | 2/21 |
| \(e_{3}\) | 6/7 | 12/7 | 18/7 | 17/7 | 16/7 | 15/7 | 2 | 13/7 | 12/7 | 11/7 | 10/7 | 9/7 | 8/7 | 1 | 6/7 | 5/7 | 4/7 | 3/7 | 2/7 | 1/7 |
| \(e_{4}\) | 17/21 | 34/21 | 17/7 | 68/21 | 64/21 | 20/7 | 8/3 | 52/21 | 16/7 | 44/21 | 40/21 | 12/7 | 32/21 | 4/3 | 8/7 | 20/21 | 16/21 | 4/7 | 8/21 | 4/21 |
| \(e_{5}\) | 16/21 | 32/21 | 16/7 | 64/21 | 80/21 | 25/7 | 10/3 | 65/21 | 20/7 | 55/21 | 50/21 | 15/7 | 40/21 | 5/3 | 10/7 | 25/21 | 20/21 | 5/7 | 10/21 | 5/21 |
| \(e_{6}\) | 5/7 | 10/7 | 15/7 | 20/7 | 25/7 | 30/7 | 4 | 26/7 | 24/7 | 22/7 | 20/7 | 18/7 | 16/7 | 2 | 12/7 | 10/7 | 8/7 | 6/7 | 4/7 | 2/7 |
| \(e_{7}\) | 2/3 | 4/3 | 2 | 8/3 | 10/3 | 4 | 14/3 | 13/3 | 4 | 11/3 | 10/3 | 3 | 8/3 | 7/3 | 2 | 5/3 | 4/3 | 1 | 2/3 | 1/3 |
| \(e_{8}\) | 13/21 | 26/21 | 13/7 | 52/21 | 65/21 | 26/7 | 13/3 | 104/21 | 32/7 | 88/21 | 80/21 | 24/7 | 64/21 | 8/3 | 16/7 | 40/21 | 32/21 | 8/7 | 16/21 | 8/21 |
| \(e_{9}\) | 4/7 | 8/7 | 12/7 | 16/7 | 20/7 | 24/7 | 4 | 32/7 | 36/7 | 33/7 | 30/7 | 27/7 | 24/7 | 3 | 18/7 | 15/7 | 12/7 | 9/7 | 6/7 | 3/7 |
| \(e_{10}\) | 11/21 | 22/21 | 11/7 | 44/21 | 55/21 | 22/7 | 11/3 | 88/21 | 33/7 | 110/21 | 100/21 | 30/7 | 80/21 | 10/3 | 20/7 | 50/21 | 40/21 | 10/7 | 20/21 | 10/21 |
| \(e_{11}\) | 10/21 | 20/21 | 10/7 | 40/21 | 50/21 | 20/7 | 10/3 | 80/21 | 30/7 | 100/21 | 110/21 | 33/7 | 88/21 | 11/3 | 22/7 | 55/21 | 44/21 | 11/7 | 22/21 | 11/21 |
| \(e_{12}\) | 3/7 | 6/7 | 9/7 | 12/7 | 15/7 | 18/7 | 3 | 24/7 | 27/7 | 30/7 | 33/7 | 36/7 | 32/7 | 4 | 24/7 | 20/7 | 16/7 | 12/7 | 8/7 | 4/7 |
| \(e_{13}\) | 8/21 | 16/21 | 8/7 | 32/21 | 40/21 | 16/7 | 8/3 | 64/21 | 24/7 | 80/21 | 88/21 | 32/7 | 104/21 | 13/3 | 26/7 | 65/21 | 52/21 | 13/7 | 26/21 | 13/21 |
| \(e_{14}\) | 1/3 | 2/3 | 1 | 4/3 | 5/3 | 2 | 7/3 | 8/3 | 3 | 10/3 | 11/3 | 4 | 13/3 | 14/3 | 4 | 10/3 | 8/3 | 2 | 4/3 | 2/3 |
| \(e_{15}\) | 2/7 | 4/7 | 6/7 | 8/7 | 10/7 | 12/7 | 2 | 16/7 | 18/7 | 20/7 | 22/7 | 24/7 | 26/7 | 4 | 30/7 | 25/7 | 20/7 | 15/7 | 10/7 | 5/7 |
| \(e_{16}\) | 5/21 | 10/21 | 5/7 | 20/21 | 25/21 | 10/7 | 5/3 | 40/21 | 15/7 | 50/21 | 55/21 | 20/7 | 65/21 | 10/3 | 25/7 | 80/21 | 64/21 | 16/7 | 32/21 | 16/21 |
| \(e_{17}\) | 4/21 | 8/21 | 4/7 | 16/21 | 20/21 | 8/7 | 4/3 | 32/21 | 12/7 | 40/21 | 44/21 | 16/7 | 52/21 | 8/3 | 20/7 | 64/21 | 68/21 | 17/7 | 34/21 | 17/21 |
| \(e_{18}\) | 1/7 | 2/7 | 3/7 | 4/7 | 5/7 | 6/7 | 1 | 8/7 | 9/7 | 10/7 | 11/7 | 12/7 | 13/7 | 2 | 15/7 | 16/7 | 17/7 | 18/7 | 12/7 | 6/7 |
| \(e_{19}\) | 2/21 | 4/21 | 2/7 | 8/21 | 10/21 | 4/7 | 2/3 | 16/21 | 6/7 | 20/21 | 22/21 | 8/7 | 26/21 | 4/3 | 10/7 | 32/21 | 34/21 | 12/7 | 38/21 | 19/21 |
| \(e_{20}\) | 1/21 | 2/21 | 1/7 | 4/21 | 5/21 | 2/7 | 1/3 | 8/21 | 3/7 | 10/21 | 11/21 | 4/7 | 13/21 | 2/3 | 5/7 | 16/21 | 17/21 | 6/7 | 19/21 | 20/21 |
[[20/21, 19/21, 6/7, 17/21, 16/21, 5/7, 2/3, 13/21, 4/7, 11/21, 10/21, 3/7, 8/21, 1/3, 2/7, 5/21, 4/21, 1/7, 2/21, 1/21], [19/21, 38/21, 12/7, 34/21, 32/21, 10/7, 4/3, 26/21, 8/7, 22/21, 20/21, 6/7, 16/21, 2/3, 4/7, 10/21, 8/21, 2/7, 4/21, 2/21], [6/7, 12/7, 18/7, 17/7, 16/7, 15/7, 2, 13/7, 12/7, 11/7, 10/7, 9/7, 8/7, 1, 6/7, 5/7, 4/7, 3/7, 2/7, 1/7], [17/21, 34/21, 17/7, 68/21, 64/21, 20/7, 8/3, 52/21, 16/7, 44/21, 40/21, 12/7, 32/21, 4/3, 8/7, 20/21, 16/21, 4/7, 8/21, 4/21], [16/21, 32/21, 16/7, 64/21, 80/21, 25/7, 10/3, 65/21, 20/7, 55/21, 50/21, 15/7, 40/21, 5/3, 10/7, 25/21, 20/21, 5/7, 10/21, 5/21], [5/7, 10/7, 15/7, 20/7, 25/7, 30/7, 4, 26/7, 24/7, 22/7, 20/7, 18/7, 16/7, 2, 12/7, 10/7, 8/7, 6/7, 4/7, 2/7], [2/3, 4/3, 2, 8/3, 10/3, 4, 14/3, 13/3, 4, 11/3, 10/3, 3, 8/3, 7/3, 2, 5/3, 4/3, 1, 2/3, 1/3], [13/21, 26/21, 13/7, 52/21, 65/21, 26/7, 13/3, 104/21, 32/7, 88/21, 80/21, 24/7, 64/21, 8/3, 16/7, 40/21, 32/21, 8/7, 16/21, 8/21], [4/7, 8/7, 12/7, 16/7, 20/7, 24/7, 4, 32/7, 36/7, 33/7, 30/7, 27/7, 24/7, 3, 18/7, 15/7, 12/7, 9/7, 6/7, 3/7], [11/21, 22/21, 11/7, 44/21, 55/21, 22/7, 11/3, 88/21, 33/7, 110/21, 100/21, 30/7, 80/21, 10/3, 20/7, 50/21, 40/21, 10/7, 20/21, 10/21], [10/21, 20/21, 10/7, 40/21, 50/21, 20/7, 10/3, 80/21, 30/7, 100/21, 110/21, 33/7, 88/21, 11/3, 22/7, 55/21, 44/21, 11/7, 22/21, 11/21], [3/7, 6/7, 9/7, 12/7, 15/7, 18/7, 3, 24/7, 27/7, 30/7, 33/7, 36/7, 32/7, 4, 24/7, 20/7, 16/7, 12/7, 8/7, 4/7], [8/21, 16/21, 8/7, 32/21, 40/21, 16/7, 8/3, 64/21, 24/7, 80/21, 88/21, 32/7, 104/21, 13/3, 26/7, 65/21, 52/21, 13/7, 26/21, 13/21], [1/3, 2/3, 1, 4/3, 5/3, 2, 7/3, 8/3, 3, 10/3, 11/3, 4, 13/3, 14/3, 4, 10/3, 8/3, 2, 4/3, 2/3], [2/7, 4/7, 6/7, 8/7, 10/7, 12/7, 2, 16/7, 18/7, 20/7, 22/7, 24/7, 26/7, 4, 30/7, 25/7, 20/7, 15/7, 10/7, 5/7], [5/21, 10/21, 5/7, 20/21, 25/21, 10/7, 5/3, 40/21, 15/7, 50/21, 55/21, 20/7, 65/21, 10/3, 25/7, 80/21, 64/21, 16/7, 32/21, 16/21], [4/21, 8/21, 4/7, 16/21, 20/21, 8/7, 4/3, 32/21, 12/7, 40/21, 44/21, 16/7, 52/21, 8/3, 20/7, 64/21, 68/21, 17/7, 34/21, 17/21], [1/7, 2/7, 3/7, 4/7, 5/7, 6/7, 1, 8/7, 9/7, 10/7, 11/7, 12/7, 13/7, 2, 15/7, 16/7, 17/7, 18/7, 12/7, 6/7], [2/21, 4/21, 2/7, 8/21, 10/21, 4/7, 2/3, 16/21, 6/7, 20/21, 22/21, 8/7, 26/21, 4/3, 10/7, 32/21, 34/21, 12/7, 38/21, 19/21], [1/21, 2/21, 1/7, 4/21, 5/21, 2/7, 1/3, 8/21, 3/7, 10/21, 11/21, 4/7, 13/21, 2/3, 5/7, 16/21, 17/21, 6/7, 19/21, 20/21]]
Definite lattice
| Minimum of \(b(x, x)\), \(x \neq 0\) | \(\tfrac{20}{21}\) |
|---|---|
| Kissing number | 42 |
| Order of the isometry group | \(102181884343418880000 = 2^{19} \cdot 3^{9} \cdot 5^{4} \cdot 7^{3} \cdot 11 \cdot 13 \cdot 17 \cdot 19\) |
Roots
| Root lattice: \(L = \mathbb{Z}\Phi(L)\) | no |
|---|---|
| Number of roots | 420 |
| \(R(L) = \mathbb{Z}\Phi(L)\) | \(\cong A_{20}\) |
| Rank of \(R(L)\) | 20 |
| \([L : R(L)]\) | 21 |
| \(R(L)\) is primitive in \(L\) | no: \(L / R(L) \cong \mathbb{Z}/21\) |
Components of \(\Phi(L)\)
| Component | Type | Values of \(b(r, r)\) | Number of roots | \(\mathbb{Z}\Phi_c\) |
|---|---|---|---|---|
| \(\Phi_{1}\) | \(A_{20}\) | \(\{2\}\) | 420 | \(\cong A_{20}\) |
Simple roots
| Simple root | Coordinates | \(b(r, r)\) |
|---|---|---|
| \(\alpha_{1}\) | \(e_{1} + e_{20}\) | \(2\) |
| \(\alpha_{2}\) | \(e_{19} - 2e_{20}\) | \(2\) |
| \(\alpha_{3}\) | \(e_{18} - 2e_{19} + e_{20}\) | \(2\) |
| \(\alpha_{4}\) | \(e_{17} - 2e_{18} + e_{19}\) | \(2\) |
| \(\alpha_{5}\) | \(e_{16} - 2e_{17} + e_{18}\) | \(2\) |
| \(\alpha_{6}\) | \(e_{15} - 2e_{16} + e_{17}\) | \(2\) |
| \(\alpha_{7}\) | \(e_{14} - 2e_{15} + e_{16}\) | \(2\) |
| \(\alpha_{8}\) | \(e_{13} - 2e_{14} + e_{15}\) | \(2\) |
| \(\alpha_{9}\) | \(e_{12} - 2e_{13} + e_{14}\) | \(2\) |
| \(\alpha_{10}\) | \(e_{11} - 2e_{12} + e_{13}\) | \(2\) |
| \(\alpha_{11}\) | \(e_{10} - 2e_{11} + e_{12}\) | \(2\) |
| \(\alpha_{12}\) | \(e_{9} - 2e_{10} + e_{11}\) | \(2\) |
| \(\alpha_{13}\) | \(e_{8} - 2e_{9} + e_{10}\) | \(2\) |
| \(\alpha_{14}\) | \(e_{7} - 2e_{8} + e_{9}\) | \(2\) |
| \(\alpha_{15}\) | \(e_{6} - 2e_{7} + e_{8}\) | \(2\) |
| \(\alpha_{16}\) | \(e_{5} - 2e_{6} + e_{7}\) | \(2\) |
| \(\alpha_{17}\) | \(e_{4} - 2e_{5} + e_{6}\) | \(2\) |
| \(\alpha_{18}\) | \(e_{3} - 2e_{4} + e_{5}\) | \(2\) |
| \(\alpha_{19}\) | \(e_{2} - 2e_{3} + e_{4}\) | \(2\) |
| \(\alpha_{20}\) | \(e_{1} - 2e_{2} + e_{3}\) | \(2\) |