Tag 0255

\(A_{20}\)

A20

Invariants

Rank	20
Signature \((n_+, n_-)\)	(20, 0)
Determinant	\(21\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{20}\).

	\(e_{1}\)	\(e_{2}\)	\(e_{3}\)	\(e_{4}\)	\(e_{5}\)	\(e_{6}\)	\(e_{7}\)	\(e_{8}\)	\(e_{9}\)	\(e_{10}\)	\(e_{11}\)	\(e_{12}\)	\(e_{13}\)	\(e_{14}\)	\(e_{15}\)	\(e_{16}\)	\(e_{17}\)	\(e_{18}\)	\(e_{19}\)	\(e_{20}\)
\(e_{1}\)	2	-1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
\(e_{2}\)	-1	2	-1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
\(e_{3}\)	0	-1	2	-1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
\(e_{4}\)	0	0	-1	2	-1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
\(e_{5}\)	0	0	0	-1	2	-1	0	0	0	0	0	0	0	0	0	0	0	0	0	0
\(e_{6}\)	0	0	0	0	-1	2	-1	0	0	0	0	0	0	0	0	0	0	0	0	0
\(e_{7}\)	0	0	0	0	0	-1	2	-1	0	0	0	0	0	0	0	0	0	0	0	0
\(e_{8}\)	0	0	0	0	0	0	-1	2	-1	0	0	0	0	0	0	0	0	0	0	0
\(e_{9}\)	0	0	0	0	0	0	0	-1	2	-1	0	0	0	0	0	0	0	0	0	0
\(e_{10}\)	0	0	0	0	0	0	0	0	-1	2	-1	0	0	0	0	0	0	0	0	0
\(e_{11}\)	0	0	0	0	0	0	0	0	0	-1	2	-1	0	0	0	0	0	0	0	0
\(e_{12}\)	0	0	0	0	0	0	0	0	0	0	-1	2	-1	0	0	0	0	0	0	0
\(e_{13}\)	0	0	0	0	0	0	0	0	0	0	0	-1	2	-1	0	0	0	0	0	0
\(e_{14}\)	0	0	0	0	0	0	0	0	0	0	0	0	-1	2	-1	0	0	0	0	0
\(e_{15}\)	0	0	0	0	0	0	0	0	0	0	0	0	0	-1	2	-1	0	0	0	0
\(e_{16}\)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	-1	2	-1	0	0	0
\(e_{17}\)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	-1	2	-1	0	0
\(e_{18}\)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	-1	2	-1	0
\(e_{19}\)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	-1	2	-1
\(e_{20}\)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	-1	2

[[2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 2, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 2, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 2, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 2, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 2, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 2, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 2, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 2, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 2]]

Integral lattice

Parity	even
Discriminant group	\(\mathbb{Z}/21\)
Unimodular	no
Integral overlattices \(L \subseteq M \subseteq L^*\)	1
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)	\(\{2, 3, 7\}\)
Character of the discriminant \(\chi_D\), \(D = (-1)^{10} \det L\)	\(\chi_{21} = \left(\frac{21}{\cdot}\right)\)
Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)	\(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 19)\, L^\Sigma(s - 10, \chi_{21})}{L^\Sigma(s - 9, \chi_{21})}\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)	\(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 19)}{L^\Sigma(s - 9, \chi_{21})}\)
Genus symbol	\(\mathrm{II}_{20,0}\ \bigl(2\colon 1^{-20};\ 3\colon 1^{-19}\,3^{-1};\ 7\colon 1^{-19}\,7^{1}\bigr)\)

Definite lattice

Minimum of \(b(x, x)\), \(x \neq 0\)	\(2\)
Kissing number	420
Order of the isometry group	\(102181884343418880000 = 2^{19} \cdot 3^{9} \cdot 5^{4} \cdot 7^{3} \cdot 11 \cdot 13 \cdot 17 \cdot 19\)

Theta series

\[ \sum_{x \in L} q^{b(x, x)} = 1 + 420q^{2} + 35910q^{4} + O(q^{5}) \]

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)	yes
\(L = \mathbb{Z}\Phi_S(L)\) for	\(S = \{2\}\)
Number of roots	420
\(R(L) = \mathbb{Z}\Phi(L)\)	\(\cong A_{20}\)
Rank of \(R(L)\)	20
\([L : R(L)]\)	1
\(R(L)\) is primitive in \(L\)	yes

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	\(b(r, r)\)
\(\alpha_{1}\)	\(e_{20}\)	\(2\)
\(\alpha_{2}\)	\(e_{19}\)	\(2\)
\(\alpha_{3}\)	\(e_{18}\)	\(2\)
\(\alpha_{4}\)	\(e_{17}\)	\(2\)
\(\alpha_{5}\)	\(e_{16}\)	\(2\)
\(\alpha_{6}\)	\(e_{15}\)	\(2\)
\(\alpha_{7}\)	\(e_{14}\)	\(2\)
\(\alpha_{8}\)	\(e_{13}\)	\(2\)
\(\alpha_{9}\)	\(e_{12}\)	\(2\)
\(\alpha_{10}\)	\(e_{11}\)	\(2\)
\(\alpha_{11}\)	\(e_{10}\)	\(2\)
\(\alpha_{12}\)	\(e_{9}\)	\(2\)
\(\alpha_{13}\)	\(e_{8}\)	\(2\)
\(\alpha_{14}\)	\(e_{7}\)	\(2\)
\(\alpha_{15}\)	\(e_{6}\)	\(2\)
\(\alpha_{16}\)	\(e_{5}\)	\(2\)
\(\alpha_{17}\)	\(e_{4}\)	\(2\)
\(\alpha_{18}\)	\(e_{3}\)	\(2\)
\(\alpha_{19}\)	\(e_{2}\)	\(2\)
\(\alpha_{20}\)	\(e_{1}\)	\(2\)