Tag 0029

\(A_{2} \oplus A_{1}\)

A2 + A1

Invariants

Rank3
Signature \((n_+, n_-)\)(3, 0)
Determinant\(6\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, e_{2}, e_{3}\).

In this basis, \(L = {}\)\(\mathbb{Z}\{e_{1}, e_{2}\}\) \(\oplus\) \(\mathbb{Z}\{e_{3}\}\), an orthogonal sum.

\(e_{1}\)\(e_{2}\)\(e_{3}\)
\(e_{1}\)2-10
\(e_{2}\)-120
\(e_{3}\)002
 
[[2, -1, 0], [-1, 2, 0], [0, 0, 2]]

Integral lattice

Parityeven
Discriminant group\(\mathbb{Z}/6\)
Unimodularno
Integral overlattices \(L \subseteq M \subseteq L^*\)1
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)\(\{2, 3\}\)
Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)\(\zeta^\Sigma(X_0, s) = \zeta^\Sigma(s - 2)\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)\(\zeta^\Sigma(X_n, s) = \zeta^\Sigma(s - 2)\, L^\Sigma(s - 1, \chi_{-6n})\)
Genus symbol\(\mathrm{II}_{3,0}\ \bigl(2\colon 1^{-2}\,[2^{1}]_{1};\ 3\colon 1^{2}\,3^{-1}\bigr)\)
Class number of the genus1

Definite lattice

Minimum of \(b(x, x)\), \(x \neq 0\)\(2\)
Kissing number8
Order of the isometry group\(24 = 2^{3} \cdot 3\)

Theta series

\[ \sum_{x \in L} q^{b(x, x)} = 1 + 8q^{2} + 12q^{4} + 6q^{6} + 20q^{8} + 24q^{10} + O(q^{13}) \]

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)yes
\(L = \mathbb{Z}\Phi_S(L)\) for\(S = \{2\}\)
Number of roots14
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong A_{2} \oplus \langle 2 \rangle\)
Rank of \(R(L)\)3
\([L : R(L)]\)1
\(R(L)\) is primitive in \(L\)yes

Components of \(\Phi(L)\)

ComponentTypeValues of \(b(r, r)\)Number of roots\(\mathbb{Z}\Phi_c\)
\(\Phi_{1}\)\(G_{2}\)\(\{2, 6\}\)12\(\cong A_{2}\)
\(\Phi_{2}\)\(A_{1}\)\(\{2\}\)2\(\cong \langle 2 \rangle\)

Simple roots

\(\alpha^{(c)}_j\) is simple root \(j\) of the component \(\Phi_c\).

Simple rootCoordinates\(b(r, r)\)
\(\alpha^{(1)}_{1}\)\(e_{2}\)\(2\)
\(\alpha^{(1)}_{2}\)\(e_{1} - e_{2}\)\(6\)
\(\alpha^{(2)}_{1}\)\(e_{3}\)\(2\)

Morphisms