Tag 0146

\(\mathrm{I}_{12,0}\)

I_{12,0}

Invariants

Rank12
Signature \((n_+, n_-)\)(12, 0)
Determinant\(1\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{12}\).

In this basis, \(L = {}\)\(\mathbb{Z}\{e_{1}\}\) \(\oplus\) \(\mathbb{Z}\{e_{2}\}\) \(\oplus\) \(\mathbb{Z}\{e_{3}\}\) \(\oplus\) \(\mathbb{Z}\{e_{4}\}\) \(\oplus\) \(\mathbb{Z}\{e_{5}\}\) \(\oplus\) \(\mathbb{Z}\{e_{6}\}\) \(\oplus\) \(\mathbb{Z}\{e_{7}\}\) \(\oplus\) \(\mathbb{Z}\{e_{8}\}\) \(\oplus\) \(\mathbb{Z}\{e_{9}\}\) \(\oplus\) \(\mathbb{Z}\{e_{10}\}\) \(\oplus\) \(\mathbb{Z}\{e_{11}\}\) \(\oplus\) \(\mathbb{Z}\{e_{12}\}\), an orthogonal sum.

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)
\(e_{1}\)100000000000
\(e_{2}\)010000000000
\(e_{3}\)001000000000
\(e_{4}\)000100000000
\(e_{5}\)000010000000
\(e_{6}\)000001000000
\(e_{7}\)000000100000
\(e_{8}\)000000010000
\(e_{9}\)000000001000
\(e_{10}\)000000000100
\(e_{11}\)000000000010
\(e_{12}\)000000000001
 
[[1, 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, 1, 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, 1, 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, 1, 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, 1, 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, 1, 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 \(\Sigma_L = \{p \mid 2 \det L\}\)\(\{2\}\)
Character of the discriminant \(\chi_D\), \(D = (-1)^{6} \det L\)trivial
Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)\(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 11)\, \zeta^\Sigma(s - 6)}{\zeta^\Sigma(s - 5)}\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)\(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 11)}{\zeta^\Sigma(s - 5)}\)
Genus symbol\(\mathrm{I}_{12,0}\)
Class number of the genus3

Definite lattice

Minimum of \(b(x, x)\), \(x \neq 0\)\(1\)
Kissing number24
Order of the isometry group\(1961990553600 = 2^{22} \cdot 3^{5} \cdot 5^{2} \cdot 7 \cdot 11\)

Theta series

\[ \sum_{x \in L} q^{b(x, x)} = 1 + 24q + 264q^{2} + 1760q^{3} + 7944q^{4} + 25872q^{5} + 64416q^{6} + O(q^{7}) \]

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)yes
\(L = \mathbb{Z}\Phi_S(L)\) for\(S = \{1\}\)
Number of roots288
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong \mathrm{I}_{12,0}\)
Rank of \(R(L)\)12
\([L : R(L)]\)1
\(R(L)\) is primitive in \(L\)yes

Components of \(\Phi(L)\)

ComponentTypeValues of \(b(r, r)\)Number of roots\(\mathbb{Z}\Phi_c\)
\(\Phi_{1}\)\(B_{12}\)\(\{1, 2\}\)288\(\cong \mathrm{I}_{12,0}\)

Simple roots

Simple rootCoordinates\(b(r, r)\)
\(\alpha_{1}\)\(e_{1} - e_{2}\)\(2\)
\(\alpha_{2}\)\(e_{2} - e_{3}\)\(2\)
\(\alpha_{3}\)\(e_{3} - e_{4}\)\(2\)
\(\alpha_{4}\)\(e_{4} - e_{5}\)\(2\)
\(\alpha_{5}\)\(e_{5} - e_{6}\)\(2\)
\(\alpha_{6}\)\(e_{6} - e_{7}\)\(2\)
\(\alpha_{7}\)\(e_{7} - e_{8}\)\(2\)
\(\alpha_{8}\)\(e_{8} - e_{9}\)\(2\)
\(\alpha_{9}\)\(e_{9} - e_{10}\)\(2\)
\(\alpha_{10}\)\(e_{10} - e_{11}\)\(2\)
\(\alpha_{11}\)\(e_{11} - e_{12}\)\(2\)
\(\alpha_{12}\)\(e_{12}\)\(1\)

Morphisms