Tag 027V

\(\mathrm{I}_{23,0}\)

I_{23,0}

Invariants

Rank23
Signature \((n_+, n_-)\)(23, 0)
Determinant\(1\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{23}\).

In this basis, \(L = {}\)\(\mathbb{Z}\{e_{1}\}\) \(\oplus\) \(\mathbb{Z}\{e_{2}\}\) \(\oplus\) \(\mathbb{Z}\{e_{3}\}\) \(\oplus\) \(\mathbb{Z}\{e_{4}\}\) \(\oplus\) \(\mathbb{Z}\{e_{5}\}\) \(\oplus\) \(\mathbb{Z}\{e_{6}\}\) \(\oplus\) \(\mathbb{Z}\{e_{7}\}\) \(\oplus\) \(\mathbb{Z}\{e_{8}\}\) \(\oplus\) \(\mathbb{Z}\{e_{9}\}\) \(\oplus\) \(\mathbb{Z}\{e_{10}\}\) \(\oplus\) \(\mathbb{Z}\{e_{11}\}\) \(\oplus\) \(\mathbb{Z}\{e_{12}\}\) \(\oplus\) \(\mathbb{Z}\{e_{13}\}\) \(\oplus\) \(\mathbb{Z}\{e_{14}\}\) \(\oplus\) \(\mathbb{Z}\{e_{15}\}\) \(\oplus\) \(\mathbb{Z}\{e_{16}\}\) \(\oplus\) \(\mathbb{Z}\{e_{17}\}\) \(\oplus\) \(\mathbb{Z}\{e_{18}\}\) \(\oplus\) \(\mathbb{Z}\{e_{19}\}\) \(\oplus\) \(\mathbb{Z}\{e_{20}\}\) \(\oplus\) \(\mathbb{Z}\{e_{21}\}\) \(\oplus\) \(\mathbb{Z}\{e_{22}\}\) \(\oplus\) \(\mathbb{Z}\{e_{23}\}\), an orthogonal sum.

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)\(e_{13}\)\(e_{14}\)\(e_{15}\)\(e_{16}\)\(e_{17}\)\(e_{18}\)\(e_{19}\)\(e_{20}\)\(e_{21}\)\(e_{22}\)\(e_{23}\)
\(e_{1}\)10000000000000000000000
\(e_{2}\)01000000000000000000000
\(e_{3}\)00100000000000000000000
\(e_{4}\)00010000000000000000000
\(e_{5}\)00001000000000000000000
\(e_{6}\)00000100000000000000000
\(e_{7}\)00000010000000000000000
\(e_{8}\)00000001000000000000000
\(e_{9}\)00000000100000000000000
\(e_{10}\)00000000010000000000000
\(e_{11}\)00000000001000000000000
\(e_{12}\)00000000000100000000000
\(e_{13}\)00000000000010000000000
\(e_{14}\)00000000000001000000000
\(e_{15}\)00000000000000100000000
\(e_{16}\)00000000000000010000000
\(e_{17}\)00000000000000001000000
\(e_{18}\)00000000000000000100000
\(e_{19}\)00000000000000000010000
\(e_{20}\)00000000000000000001000
\(e_{21}\)00000000000000000000100
\(e_{22}\)00000000000000000000010
\(e_{23}\)00000000000000000000001
 
[[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]]

Integral lattice

Parityodd
Discriminant group\(0\)
Unimodularyes
Integral overlattices \(L \subseteq M \subseteq L^*\)1
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)\(\{2\}\)
Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)\(\zeta^\Sigma(X_0, s) = \zeta^\Sigma(s - 22)\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)\(\zeta^\Sigma(X_n, s) = \zeta^\Sigma(s - 22)\, L^\Sigma(s - 11, \chi_{-n})\)

Definite lattice

Minimum of \(b(x, x)\), \(x \neq 0\)\(1\)
Kissing number46

Theta series

\[ \sum_{x \in L} q^{b(x, x)} = 1 + 46q + 1012q^{2} + 14168q^{3} + 141726q^{4} + O(q^{5}) \]

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)yes
\(L = \mathbb{Z}\Phi_S(L)\) for\(S = \{1\}\)
Number of roots1058
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong \mathrm{I}_{23,0}\)
Rank of \(R(L)\)23
\([L : R(L)]\)1
\(R(L)\) is primitive in \(L\)yes

Components of \(\Phi(L)\)

ComponentTypeValues of \(b(r, r)\)Number of roots\(\mathbb{Z}\Phi_c\)
\(\Phi_{1}\)\(B_{23}\)\(\{1, 2\}\)1058\(\cong \mathrm{I}_{23,0}\)

Simple roots

Simple rootCoordinates\(b(r, r)\)
\(\alpha_{1}\)\(e_{1} - e_{2}\)\(2\)
\(\alpha_{2}\)\(e_{2} - e_{3}\)\(2\)
\(\alpha_{3}\)\(e_{3} - e_{4}\)\(2\)
\(\alpha_{4}\)\(e_{4} - e_{5}\)\(2\)
\(\alpha_{5}\)\(e_{5} - e_{6}\)\(2\)
\(\alpha_{6}\)\(e_{6} - e_{7}\)\(2\)
\(\alpha_{7}\)\(e_{7} - e_{8}\)\(2\)
\(\alpha_{8}\)\(e_{8} - e_{9}\)\(2\)
\(\alpha_{9}\)\(e_{9} - e_{10}\)\(2\)
\(\alpha_{10}\)\(e_{10} - e_{11}\)\(2\)
\(\alpha_{11}\)\(e_{11} - e_{12}\)\(2\)
\(\alpha_{12}\)\(e_{12} - e_{13}\)\(2\)
\(\alpha_{13}\)\(e_{13} - e_{14}\)\(2\)
\(\alpha_{14}\)\(e_{14} - e_{15}\)\(2\)
\(\alpha_{15}\)\(e_{15} - e_{16}\)\(2\)
\(\alpha_{16}\)\(e_{16} - e_{17}\)\(2\)
\(\alpha_{17}\)\(e_{17} - e_{18}\)\(2\)
\(\alpha_{18}\)\(e_{18} - e_{19}\)\(2\)
\(\alpha_{19}\)\(e_{19} - e_{20}\)\(2\)
\(\alpha_{20}\)\(e_{20} - e_{21}\)\(2\)
\(\alpha_{21}\)\(e_{21} - e_{22}\)\(2\)
\(\alpha_{22}\)\(e_{22} - e_{23}\)\(2\)
\(\alpha_{23}\)\(e_{23}\)\(1\)

Morphisms