Tag 0011

\(\mathrm{I}_{2,0}\)

I_{2,0}

Invariants

Rank2
Signature \((n_+, n_-)\)(2, 0)
Determinant\(1\)

Gram tensor

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

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

\(e_{1}\)\(e_{2}\)
\(e_{1}\)10
\(e_{2}\)01
 
[[1, 0], [0, 1]]

Integral lattice

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

Definite lattice

Minimum of \(b(x, x)\), \(x \neq 0\)\(1\)
Kissing number4
Order of the isometry group\(8 = 2^{3}\)

Theta series

\[ \sum_{x \in L} q^{b(x, x)} = 1 + 4q + 4q^{2} + 4q^{4} + 8q^{5} + 4q^{8} + 4q^{9} + 8q^{10} + O(q^{13}) \]

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)yes
\(L = \mathbb{Z}\Phi_S(L)\) for\(S = \{1\}\)
Number of roots8
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong \mathrm{I}_{2,0}\)
Rank of \(R(L)\)2
\([L : R(L)]\)1
\(R(L)\) is primitive in \(L\)yes

Components of \(\Phi(L)\)

ComponentTypeValues of \(b(r, r)\)Number of roots\(\mathbb{Z}\Phi_c\)
\(\Phi_{1}\)\(B_{2}\)\(\{1, 2\}\)8\(\cong \mathrm{I}_{2,0}\)

Simple roots

Simple rootCoordinates\(b(r, r)\)
\(\alpha_{1}\)\(e_{1} - e_{2}\)\(2\)
\(\alpha_{2}\)\(e_{2}\)\(1\)

Morphisms