Tag 0190

\(\mathrm{I}_{15,0}\)

I_{15,0}

Invariants

Rank15
Signature \((n_+, n_-)\)(15, 0)
Determinant\(1\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{15}\).

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

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)\(e_{13}\)\(e_{14}\)\(e_{15}\)
\(e_{1}\)100000000000000
\(e_{2}\)010000000000000
\(e_{3}\)001000000000000
\(e_{4}\)000100000000000
\(e_{5}\)000010000000000
\(e_{6}\)000001000000000
\(e_{7}\)000000100000000
\(e_{8}\)000000010000000
\(e_{9}\)000000001000000
\(e_{10}\)000000000100000
\(e_{11}\)000000000010000
\(e_{12}\)000000000001000
\(e_{13}\)000000000000100
\(e_{14}\)000000000000010
\(e_{15}\)000000000000001
 
[[1, 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, 1, 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, 1, 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, 1, 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, 1, 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, 1, 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, 1, 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, 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 \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)\(\zeta^\Sigma(X_0, s) = \zeta^\Sigma(s - 14)\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)\(\zeta^\Sigma(X_n, s) = \zeta^\Sigma(s - 14)\, L^\Sigma(s - 7, \chi_{-n})\)
Genus symbol\(\mathrm{I}_{15,0}\)

Definite lattice

Minimum of \(b(x, x)\), \(x \neq 0\)\(1\)
Kissing number30
Order of the isometry group\(42849873690624000 = 2^{26} \cdot 3^{6} \cdot 5^{3} \cdot 7^{2} \cdot 11 \cdot 13\)

Theta series

\[ \sum_{x \in L} q^{b(x, x)} = 1 + 30q + 420q^{2} + 3640q^{3} + 21870q^{4} + O(q^{5}) \]

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)yes
\(L = \mathbb{Z}\Phi_S(L)\) for\(S = \{1\}\)
Number of roots450
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong \mathrm{I}_{15,0}\)
Rank of \(R(L)\)15
\([L : R(L)]\)1
\(R(L)\) is primitive in \(L\)yes

Components of \(\Phi(L)\)

ComponentTypeValues of \(b(r, r)\)Number of roots\(\mathbb{Z}\Phi_c\)
\(\Phi_{1}\)\(B_{15}\)\(\{1, 2\}\)450\(\cong \mathrm{I}_{15,0}\)

Simple roots

Simple rootCoordinates\(b(r, r)\)
\(\alpha_{1}\)\(e_{1} - e_{2}\)\(2\)
\(\alpha_{2}\)\(e_{2} - e_{3}\)\(2\)
\(\alpha_{3}\)\(e_{3} - e_{4}\)\(2\)
\(\alpha_{4}\)\(e_{4} - e_{5}\)\(2\)
\(\alpha_{5}\)\(e_{5} - e_{6}\)\(2\)
\(\alpha_{6}\)\(e_{6} - e_{7}\)\(2\)
\(\alpha_{7}\)\(e_{7} - e_{8}\)\(2\)
\(\alpha_{8}\)\(e_{8} - e_{9}\)\(2\)
\(\alpha_{9}\)\(e_{9} - e_{10}\)\(2\)
\(\alpha_{10}\)\(e_{10} - e_{11}\)\(2\)
\(\alpha_{11}\)\(e_{11} - e_{12}\)\(2\)
\(\alpha_{12}\)\(e_{12} - e_{13}\)\(2\)
\(\alpha_{13}\)\(e_{13} - e_{14}\)\(2\)
\(\alpha_{14}\)\(e_{14} - e_{15}\)\(2\)
\(\alpha_{15}\)\(e_{15}\)\(1\)

Morphisms