Tag 0237

\(A_{18}^{*}\)

A18*

Invariants

Rank	18
Signature \((n_+, n_-)\)	(18, 0)
Determinant	\(\tfrac{1}{19}\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{18}\).

	\(e_{1}\)	\(e_{2}\)	\(e_{3}\)	\(e_{4}\)	\(e_{5}\)	\(e_{6}\)	\(e_{7}\)	\(e_{8}\)	\(e_{9}\)	\(e_{10}\)	\(e_{11}\)	\(e_{12}\)	\(e_{13}\)	\(e_{14}\)	\(e_{15}\)	\(e_{16}\)	\(e_{17}\)	\(e_{18}\)
\(e_{1}\)	18/19	17/19	16/19	15/19	14/19	13/19	12/19	11/19	10/19	9/19	8/19	7/19	6/19	5/19	4/19	3/19	2/19	1/19
\(e_{2}\)	17/19	34/19	32/19	30/19	28/19	26/19	24/19	22/19	20/19	18/19	16/19	14/19	12/19	10/19	8/19	6/19	4/19	2/19
\(e_{3}\)	16/19	32/19	48/19	45/19	42/19	39/19	36/19	33/19	30/19	27/19	24/19	21/19	18/19	15/19	12/19	9/19	6/19	3/19
\(e_{4}\)	15/19	30/19	45/19	60/19	56/19	52/19	48/19	44/19	40/19	36/19	32/19	28/19	24/19	20/19	16/19	12/19	8/19	4/19
\(e_{5}\)	14/19	28/19	42/19	56/19	70/19	65/19	60/19	55/19	50/19	45/19	40/19	35/19	30/19	25/19	20/19	15/19	10/19	5/19
\(e_{6}\)	13/19	26/19	39/19	52/19	65/19	78/19	72/19	66/19	60/19	54/19	48/19	42/19	36/19	30/19	24/19	18/19	12/19	6/19
\(e_{7}\)	12/19	24/19	36/19	48/19	60/19	72/19	84/19	77/19	70/19	63/19	56/19	49/19	42/19	35/19	28/19	21/19	14/19	7/19
\(e_{8}\)	11/19	22/19	33/19	44/19	55/19	66/19	77/19	88/19	80/19	72/19	64/19	56/19	48/19	40/19	32/19	24/19	16/19	8/19
\(e_{9}\)	10/19	20/19	30/19	40/19	50/19	60/19	70/19	80/19	90/19	81/19	72/19	63/19	54/19	45/19	36/19	27/19	18/19	9/19
\(e_{10}\)	9/19	18/19	27/19	36/19	45/19	54/19	63/19	72/19	81/19	90/19	80/19	70/19	60/19	50/19	40/19	30/19	20/19	10/19
\(e_{11}\)	8/19	16/19	24/19	32/19	40/19	48/19	56/19	64/19	72/19	80/19	88/19	77/19	66/19	55/19	44/19	33/19	22/19	11/19
\(e_{12}\)	7/19	14/19	21/19	28/19	35/19	42/19	49/19	56/19	63/19	70/19	77/19	84/19	72/19	60/19	48/19	36/19	24/19	12/19
\(e_{13}\)	6/19	12/19	18/19	24/19	30/19	36/19	42/19	48/19	54/19	60/19	66/19	72/19	78/19	65/19	52/19	39/19	26/19	13/19
\(e_{14}\)	5/19	10/19	15/19	20/19	25/19	30/19	35/19	40/19	45/19	50/19	55/19	60/19	65/19	70/19	56/19	42/19	28/19	14/19
\(e_{15}\)	4/19	8/19	12/19	16/19	20/19	24/19	28/19	32/19	36/19	40/19	44/19	48/19	52/19	56/19	60/19	45/19	30/19	15/19
\(e_{16}\)	3/19	6/19	9/19	12/19	15/19	18/19	21/19	24/19	27/19	30/19	33/19	36/19	39/19	42/19	45/19	48/19	32/19	16/19
\(e_{17}\)	2/19	4/19	6/19	8/19	10/19	12/19	14/19	16/19	18/19	20/19	22/19	24/19	26/19	28/19	30/19	32/19	34/19	17/19
\(e_{18}\)	1/19	2/19	3/19	4/19	5/19	6/19	7/19	8/19	9/19	10/19	11/19	12/19	13/19	14/19	15/19	16/19	17/19	18/19

[[18/19, 17/19, 16/19, 15/19, 14/19, 13/19, 12/19, 11/19, 10/19, 9/19, 8/19, 7/19, 6/19, 5/19, 4/19, 3/19, 2/19, 1/19], [17/19, 34/19, 32/19, 30/19, 28/19, 26/19, 24/19, 22/19, 20/19, 18/19, 16/19, 14/19, 12/19, 10/19, 8/19, 6/19, 4/19, 2/19], [16/19, 32/19, 48/19, 45/19, 42/19, 39/19, 36/19, 33/19, 30/19, 27/19, 24/19, 21/19, 18/19, 15/19, 12/19, 9/19, 6/19, 3/19], [15/19, 30/19, 45/19, 60/19, 56/19, 52/19, 48/19, 44/19, 40/19, 36/19, 32/19, 28/19, 24/19, 20/19, 16/19, 12/19, 8/19, 4/19], [14/19, 28/19, 42/19, 56/19, 70/19, 65/19, 60/19, 55/19, 50/19, 45/19, 40/19, 35/19, 30/19, 25/19, 20/19, 15/19, 10/19, 5/19], [13/19, 26/19, 39/19, 52/19, 65/19, 78/19, 72/19, 66/19, 60/19, 54/19, 48/19, 42/19, 36/19, 30/19, 24/19, 18/19, 12/19, 6/19], [12/19, 24/19, 36/19, 48/19, 60/19, 72/19, 84/19, 77/19, 70/19, 63/19, 56/19, 49/19, 42/19, 35/19, 28/19, 21/19, 14/19, 7/19], [11/19, 22/19, 33/19, 44/19, 55/19, 66/19, 77/19, 88/19, 80/19, 72/19, 64/19, 56/19, 48/19, 40/19, 32/19, 24/19, 16/19, 8/19], [10/19, 20/19, 30/19, 40/19, 50/19, 60/19, 70/19, 80/19, 90/19, 81/19, 72/19, 63/19, 54/19, 45/19, 36/19, 27/19, 18/19, 9/19], [9/19, 18/19, 27/19, 36/19, 45/19, 54/19, 63/19, 72/19, 81/19, 90/19, 80/19, 70/19, 60/19, 50/19, 40/19, 30/19, 20/19, 10/19], [8/19, 16/19, 24/19, 32/19, 40/19, 48/19, 56/19, 64/19, 72/19, 80/19, 88/19, 77/19, 66/19, 55/19, 44/19, 33/19, 22/19, 11/19], [7/19, 14/19, 21/19, 28/19, 35/19, 42/19, 49/19, 56/19, 63/19, 70/19, 77/19, 84/19, 72/19, 60/19, 48/19, 36/19, 24/19, 12/19], [6/19, 12/19, 18/19, 24/19, 30/19, 36/19, 42/19, 48/19, 54/19, 60/19, 66/19, 72/19, 78/19, 65/19, 52/19, 39/19, 26/19, 13/19], [5/19, 10/19, 15/19, 20/19, 25/19, 30/19, 35/19, 40/19, 45/19, 50/19, 55/19, 60/19, 65/19, 70/19, 56/19, 42/19, 28/19, 14/19], [4/19, 8/19, 12/19, 16/19, 20/19, 24/19, 28/19, 32/19, 36/19, 40/19, 44/19, 48/19, 52/19, 56/19, 60/19, 45/19, 30/19, 15/19], [3/19, 6/19, 9/19, 12/19, 15/19, 18/19, 21/19, 24/19, 27/19, 30/19, 33/19, 36/19, 39/19, 42/19, 45/19, 48/19, 32/19, 16/19], [2/19, 4/19, 6/19, 8/19, 10/19, 12/19, 14/19, 16/19, 18/19, 20/19, 22/19, 24/19, 26/19, 28/19, 30/19, 32/19, 34/19, 17/19], [1/19, 2/19, 3/19, 4/19, 5/19, 6/19, 7/19, 8/19, 9/19, 10/19, 11/19, 12/19, 13/19, 14/19, 15/19, 16/19, 17/19, 18/19]]

Definite lattice

Minimum of \(b(x, x)\), \(x \neq 0\)	\(\tfrac{18}{19}\)
Kissing number	38
Order of the isometry group	\(243290200817664000 = 2^{17} \cdot 3^{8} \cdot 5^{3} \cdot 7^{2} \cdot 11 \cdot 13 \cdot 17 \cdot 19\)

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)	no
Number of roots	342
\(R(L) = \mathbb{Z}\Phi(L)\)	\(\cong A_{18}\)
Rank of \(R(L)\)	18
\([L : R(L)]\)	19
\(R(L)\) is primitive in \(L\)	no: \(L / R(L) \cong \mathbb{Z}/19\)

Components of \(\Phi(L)\)

Component	Type	Values of \(b(r, r)\)	Number of roots	\(\mathbb{Z}\Phi_c\)
\(\Phi_{1}\)	\(A_{18}\)	\(\{2\}\)	342	\(\cong A_{18}\)

Simple roots

Simple root	Coordinates	\(b(r, r)\)
\(\alpha_{1}\)	\(e_{1} + e_{18}\)	\(2\)
\(\alpha_{2}\)	\(e_{17} - 2e_{18}\)	\(2\)
\(\alpha_{3}\)	\(e_{16} - 2e_{17} + e_{18}\)	\(2\)
\(\alpha_{4}\)	\(e_{15} - 2e_{16} + e_{17}\)	\(2\)
\(\alpha_{5}\)	\(e_{14} - 2e_{15} + e_{16}\)	\(2\)
\(\alpha_{6}\)	\(e_{13} - 2e_{14} + e_{15}\)	\(2\)
\(\alpha_{7}\)	\(e_{12} - 2e_{13} + e_{14}\)	\(2\)
\(\alpha_{8}\)	\(e_{11} - 2e_{12} + e_{13}\)	\(2\)
\(\alpha_{9}\)	\(e_{10} - 2e_{11} + e_{12}\)	\(2\)
\(\alpha_{10}\)	\(e_{9} - 2e_{10} + e_{11}\)	\(2\)
\(\alpha_{11}\)	\(e_{8} - 2e_{9} + e_{10}\)	\(2\)
\(\alpha_{12}\)	\(e_{7} - 2e_{8} + e_{9}\)	\(2\)
\(\alpha_{13}\)	\(e_{6} - 2e_{7} + e_{8}\)	\(2\)
\(\alpha_{14}\)	\(e_{5} - 2e_{6} + e_{7}\)	\(2\)
\(\alpha_{15}\)	\(e_{4} - 2e_{5} + e_{6}\)	\(2\)
\(\alpha_{16}\)	\(e_{3} - 2e_{4} + e_{5}\)	\(2\)
\(\alpha_{17}\)	\(e_{2} - 2e_{3} + e_{4}\)	\(2\)
\(\alpha_{18}\)	\(e_{1} - 2e_{2} + e_{3}\)	\(2\)