Tag 0249

\(A_{19}^{*}\)

A19*

Invariants

Rank19
Signature \((n_+, n_-)\)(19, 0)
Determinant\(\tfrac{1}{20}\)

Gram tensor

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

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)\(e_{13}\)\(e_{14}\)\(e_{15}\)\(e_{16}\)\(e_{17}\)\(e_{18}\)\(e_{19}\)
\(e_{1}\)19/209/1017/204/53/47/1013/203/511/201/29/202/57/203/101/41/53/201/101/20
\(e_{2}\)9/109/517/108/53/27/513/106/511/1019/104/57/103/51/22/53/101/51/10
\(e_{3}\)17/2017/1051/2012/59/421/1039/209/533/203/227/206/521/209/103/43/59/203/103/20
\(e_{4}\)4/58/512/516/5314/513/512/511/529/58/57/56/514/53/52/51/5
\(e_{5}\)3/43/29/4315/47/213/4311/45/29/427/43/25/413/41/21/4
\(e_{6}\)7/107/521/1014/57/221/539/1018/533/10327/1012/521/109/53/26/59/103/53/10
\(e_{7}\)13/2013/1039/2013/513/439/1091/2021/577/207/263/2014/549/2021/107/47/521/207/107/20
\(e_{8}\)3/56/59/512/5318/521/524/522/5418/516/514/512/528/56/54/52/5
\(e_{9}\)11/2011/1033/2011/511/433/1077/2022/599/209/281/2018/563/2027/109/49/527/209/109/20
\(e_{10}\)1/213/225/237/249/259/247/235/223/211/2
\(e_{11}\)9/209/1027/209/59/427/1063/2018/581/209/299/2022/577/2033/1011/411/533/2011/1011/20
\(e_{12}\)2/54/56/58/5212/514/516/518/5422/524/521/518/5312/59/56/53/5
\(e_{13}\)7/207/1021/207/57/421/1049/2014/563/207/277/2021/591/2039/1013/413/539/2013/1013/20
\(e_{14}\)3/103/59/106/53/29/521/1012/527/10333/1018/539/1021/57/214/521/107/57/10
\(e_{15}\)1/41/23/415/43/27/429/45/211/4313/47/215/439/43/23/4
\(e_{16}\)1/52/53/54/516/57/58/59/5211/512/513/514/5316/512/58/54/5
\(e_{17}\)3/203/109/203/53/49/1021/206/527/203/233/209/539/2021/109/412/551/2017/1017/20
\(e_{18}\)1/101/53/102/51/23/57/104/59/10111/106/513/107/53/28/517/109/59/10
\(e_{19}\)1/201/103/201/51/43/107/202/59/201/211/203/513/207/103/44/517/209/1019/20
 
[[19/20, 9/10, 17/20, 4/5, 3/4, 7/10, 13/20, 3/5, 11/20, 1/2, 9/20, 2/5, 7/20, 3/10, 1/4, 1/5, 3/20, 1/10, 1/20], [9/10, 9/5, 17/10, 8/5, 3/2, 7/5, 13/10, 6/5, 11/10, 1, 9/10, 4/5, 7/10, 3/5, 1/2, 2/5, 3/10, 1/5, 1/10], [17/20, 17/10, 51/20, 12/5, 9/4, 21/10, 39/20, 9/5, 33/20, 3/2, 27/20, 6/5, 21/20, 9/10, 3/4, 3/5, 9/20, 3/10, 3/20], [4/5, 8/5, 12/5, 16/5, 3, 14/5, 13/5, 12/5, 11/5, 2, 9/5, 8/5, 7/5, 6/5, 1, 4/5, 3/5, 2/5, 1/5], [3/4, 3/2, 9/4, 3, 15/4, 7/2, 13/4, 3, 11/4, 5/2, 9/4, 2, 7/4, 3/2, 5/4, 1, 3/4, 1/2, 1/4], [7/10, 7/5, 21/10, 14/5, 7/2, 21/5, 39/10, 18/5, 33/10, 3, 27/10, 12/5, 21/10, 9/5, 3/2, 6/5, 9/10, 3/5, 3/10], [13/20, 13/10, 39/20, 13/5, 13/4, 39/10, 91/20, 21/5, 77/20, 7/2, 63/20, 14/5, 49/20, 21/10, 7/4, 7/5, 21/20, 7/10, 7/20], [3/5, 6/5, 9/5, 12/5, 3, 18/5, 21/5, 24/5, 22/5, 4, 18/5, 16/5, 14/5, 12/5, 2, 8/5, 6/5, 4/5, 2/5], [11/20, 11/10, 33/20, 11/5, 11/4, 33/10, 77/20, 22/5, 99/20, 9/2, 81/20, 18/5, 63/20, 27/10, 9/4, 9/5, 27/20, 9/10, 9/20], [1/2, 1, 3/2, 2, 5/2, 3, 7/2, 4, 9/2, 5, 9/2, 4, 7/2, 3, 5/2, 2, 3/2, 1, 1/2], [9/20, 9/10, 27/20, 9/5, 9/4, 27/10, 63/20, 18/5, 81/20, 9/2, 99/20, 22/5, 77/20, 33/10, 11/4, 11/5, 33/20, 11/10, 11/20], [2/5, 4/5, 6/5, 8/5, 2, 12/5, 14/5, 16/5, 18/5, 4, 22/5, 24/5, 21/5, 18/5, 3, 12/5, 9/5, 6/5, 3/5], [7/20, 7/10, 21/20, 7/5, 7/4, 21/10, 49/20, 14/5, 63/20, 7/2, 77/20, 21/5, 91/20, 39/10, 13/4, 13/5, 39/20, 13/10, 13/20], [3/10, 3/5, 9/10, 6/5, 3/2, 9/5, 21/10, 12/5, 27/10, 3, 33/10, 18/5, 39/10, 21/5, 7/2, 14/5, 21/10, 7/5, 7/10], [1/4, 1/2, 3/4, 1, 5/4, 3/2, 7/4, 2, 9/4, 5/2, 11/4, 3, 13/4, 7/2, 15/4, 3, 9/4, 3/2, 3/4], [1/5, 2/5, 3/5, 4/5, 1, 6/5, 7/5, 8/5, 9/5, 2, 11/5, 12/5, 13/5, 14/5, 3, 16/5, 12/5, 8/5, 4/5], [3/20, 3/10, 9/20, 3/5, 3/4, 9/10, 21/20, 6/5, 27/20, 3/2, 33/20, 9/5, 39/20, 21/10, 9/4, 12/5, 51/20, 17/10, 17/20], [1/10, 1/5, 3/10, 2/5, 1/2, 3/5, 7/10, 4/5, 9/10, 1, 11/10, 6/5, 13/10, 7/5, 3/2, 8/5, 17/10, 9/5, 9/10], [1/20, 1/10, 3/20, 1/5, 1/4, 3/10, 7/20, 2/5, 9/20, 1/2, 11/20, 3/5, 13/20, 7/10, 3/4, 4/5, 17/20, 9/10, 19/20]]

Definite lattice

Minimum of \(b(x, x)\), \(x \neq 0\)\(\tfrac{19}{20}\)
Kissing number40
Order of the isometry group\(4865804016353280000 = 2^{19} \cdot 3^{8} \cdot 5^{4} \cdot 7^{2} \cdot 11 \cdot 13 \cdot 17 \cdot 19\)

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)no
Number of roots380
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong A_{19}\)
Rank of \(R(L)\)19
\([L : R(L)]\)20
\(R(L)\) is primitive in \(L\)no: \(L / R(L) \cong \mathbb{Z}/20\)

Components of \(\Phi(L)\)

ComponentTypeValues of \(b(r, r)\)Number of roots\(\mathbb{Z}\Phi_c\)
\(\Phi_{1}\)\(A_{19}\)\(\{2\}\)380\(\cong A_{19}\)

Simple roots

Simple rootCoordinates\(b(r, r)\)
\(\alpha_{1}\)\(e_{1} + e_{19}\)\(2\)
\(\alpha_{2}\)\(e_{18} - 2e_{19}\)\(2\)
\(\alpha_{3}\)\(e_{17} - 2e_{18} + e_{19}\)\(2\)
\(\alpha_{4}\)\(e_{16} - 2e_{17} + e_{18}\)\(2\)
\(\alpha_{5}\)\(e_{15} - 2e_{16} + e_{17}\)\(2\)
\(\alpha_{6}\)\(e_{14} - 2e_{15} + e_{16}\)\(2\)
\(\alpha_{7}\)\(e_{13} - 2e_{14} + e_{15}\)\(2\)
\(\alpha_{8}\)\(e_{12} - 2e_{13} + e_{14}\)\(2\)
\(\alpha_{9}\)\(e_{11} - 2e_{12} + e_{13}\)\(2\)
\(\alpha_{10}\)\(e_{10} - 2e_{11} + e_{12}\)\(2\)
\(\alpha_{11}\)\(e_{9} - 2e_{10} + e_{11}\)\(2\)
\(\alpha_{12}\)\(e_{8} - 2e_{9} + e_{10}\)\(2\)
\(\alpha_{13}\)\(e_{7} - 2e_{8} + e_{9}\)\(2\)
\(\alpha_{14}\)\(e_{6} - 2e_{7} + e_{8}\)\(2\)
\(\alpha_{15}\)\(e_{5} - 2e_{6} + e_{7}\)\(2\)
\(\alpha_{16}\)\(e_{4} - 2e_{5} + e_{6}\)\(2\)
\(\alpha_{17}\)\(e_{3} - 2e_{4} + e_{5}\)\(2\)
\(\alpha_{18}\)\(e_{2} - 2e_{3} + e_{4}\)\(2\)
\(\alpha_{19}\)\(e_{1} - 2e_{2} + e_{3}\)\(2\)
  • 0243 \(A_{19}\) The dual lattice \(L^*\).