Tag 0001

\(\langle 1 \rangle\)

<1>·I_{1,0}

Invariants

Rank1
Signature \((n_+, n_-)\)(1, 0)
Determinant\(1\)

Gram tensor

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

\(e_{1}\)
\(e_{1}\)1
 
[[1]]

Integral lattice

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

Definite lattice

Minimum of \(b(x, x)\), \(x \neq 0\)\(1\)
Kissing number2
Order of the isometry group\(2 = 2\)

Theta series

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

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)yes
\(L = \mathbb{Z}\Phi_S(L)\) for\(S = \{1\}\)
Number of roots2
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong \langle 1 \rangle\)
Rank of \(R(L)\)1
\([L : R(L)]\)1
\(R(L)\) is primitive in \(L\)yes

Components of \(\Phi(L)\)

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

Simple roots

Simple rootCoordinates\(b(r, r)\)
\(\alpha_{1}\)\(e_{1}\)\(1\)
  • 0006 \(\langle \frac{1}{2} \rangle\) The same module with the form \(\tfrac{1}{2} b\).
  • 0002 \(\langle 2 \rangle\) The same module with the form \(2b\).

Morphisms