Tag 0034

\(A_{4}\)

A4

Invariants

Rank4
Signature \((n_+, n_-)\)(4, 0)
Determinant\(5\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{4}\).

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)
\(e_{1}\)2-100
\(e_{2}\)-12-10
\(e_{3}\)0-12-1
\(e_{4}\)00-12
 
[[2, -1, 0, 0], [-1, 2, -1, 0], [0, -1, 2, -1], [0, 0, -1, 2]]

Integral lattice

Parityeven
Discriminant group\(\mathbb{Z}/5\)
Unimodularno
Integral overlattices \(L \subseteq M \subseteq L^*\)1
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)\(\{2, 5\}\)
Character of the discriminant \(\chi_D\), \(D = (-1)^{2} \det L\)\(\chi_{5} = \left(\frac{5}{\cdot}\right)\)
Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)\(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 3)\, L^\Sigma(s - 2, \chi_{5})}{L^\Sigma(s - 1, \chi_{5})}\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)\(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 3)}{L^\Sigma(s - 1, \chi_{5})}\)
Genus symbol\(\mathrm{II}_{4,0}\ \bigl(2\colon 1^{-4};\ 5\colon 1^{3}\,5^{1}\bigr)\)
Class number of the genus1

Definite lattice

Minimum of \(b(x, x)\), \(x \neq 0\)\(2\)
Kissing number20
Order of the isometry group\(240 = 2^{4} \cdot 3 \cdot 5\)

Theta series

\[ \sum_{x \in L} q^{b(x, x)} = 1 + 20q^{2} + 30q^{4} + 60q^{6} + 60q^{8} + 120q^{10} + 40q^{12} + O(q^{13}) \]

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)yes
\(L = \mathbb{Z}\Phi_S(L)\) for\(S = \{2\}\)
Number of roots20
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong A_{4}\)
Rank of \(R(L)\)4
\([L : R(L)]\)1
\(R(L)\) is primitive in \(L\)yes

Components of \(\Phi(L)\)

ComponentTypeValues of \(b(r, r)\)Number of roots\(\mathbb{Z}\Phi_c\)
\(\Phi_{1}\)\(A_{4}\)\(\{2\}\)20\(\cong A_{4}\)

Simple roots

Simple rootCoordinates\(b(r, r)\)
\(\alpha_{1}\)\(e_{4}\)\(2\)
\(\alpha_{2}\)\(e_{3}\)\(2\)
\(\alpha_{3}\)\(e_{2}\)\(2\)
\(\alpha_{4}\)\(e_{1}\)\(2\)
  • 0041 \(A_{4}^{*}\) The dual lattice \(L^*\).