Tag 0110

\(\Lambda_{9}\)

Lambda9·LAMBDA9

Invariants

Rank9
Signature \((n_+, n_-)\)(9, 0)
Determinant\(512\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{9}\).

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)
\(e_{1}\)4-20000000
\(e_{2}\)-24-2200000
\(e_{3}\)0-24002000
\(e_{4}\)020422000
\(e_{5}\)000242002
\(e_{6}\)002224221
\(e_{7}\)000002420
\(e_{8}\)000002240
\(e_{9}\)000021004
 
[[4, -2, 0, 0, 0, 0, 0, 0, 0], [-2, 4, -2, 2, 0, 0, 0, 0, 0], [0, -2, 4, 0, 0, 2, 0, 0, 0], [0, 2, 0, 4, 2, 2, 0, 0, 0], [0, 0, 0, 2, 4, 2, 0, 0, 2], [0, 0, 2, 2, 2, 4, 2, 2, 1], [0, 0, 0, 0, 0, 2, 4, 2, 0], [0, 0, 0, 0, 0, 2, 2, 4, 0], [0, 0, 0, 0, 2, 1, 0, 0, 4]]

Integral lattice

Parityeven
Discriminant group\((\mathbb{Z}/2)^{6} \oplus \mathbb{Z}/8\)
Unimodularno
Integral overlattices \(L \subseteq M \subseteq L^*\)2981
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)\(\{2\}\)
Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)\(\zeta^\Sigma(X_0, s) = \zeta^\Sigma(s - 8)\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)\(\zeta^\Sigma(X_n, s) = \zeta^\Sigma(s - 8)\, L^\Sigma(s - 4, \chi_{512n})\)
Genus symbol\(\mathrm{II}_{9,0}\ \bigl(2\colon 1^{2}:2^{6}:[8^{1}]_{1}\bigr)\)
Class number of the genus3

Definite lattice

Minimum of \(b(x, x)\), \(x \neq 0\)\(4\)
Kissing number272
Order of the isometry group\(10321920 = 2^{15} \cdot 3^{2} \cdot 5 \cdot 7\)

Theta series

\[ \sum_{x \in L} q^{b(x, x)} = 1 + 272q^{4} + 256q^{6} + O(q^{7}) \]

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)no
Number of roots114
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong D_{8}(2) \oplus \langle 8 \rangle\)
Rank of \(R(L)\)9
\([L : R(L)]\)4
\(R(L)\) is primitive in \(L\)no: \(L / R(L) \cong (\mathbb{Z}/2)^{2}\)

Components of \(\Phi(L)\)

ComponentTypeValues of \(b(r, r)\)Number of roots\(\mathbb{Z}\Phi_c\)
\(\Phi_{1}\)\(D_{8}\)\(\{4\}\)112\(\cong D_{8}(2)\)
\(\Phi_{2}\)\(A_{1}\)\(\{8\}\)2\(\cong \langle 8 \rangle\)

Simple roots

\(\alpha^{(c)}_j\) is simple root \(j\) of the component \(\Phi_c\).

Simple rootCoordinates\(b(r, r)\)
\(\alpha^{(1)}_{1}\)\(e_{8}\)\(4\)
\(\alpha^{(1)}_{2}\)\(e_{7} - e_{8}\)\(4\)
\(\alpha^{(1)}_{3}\)\(e_{3} + e_{5} - 2e_{6} + e_{8}\)\(4\)
\(\alpha^{(1)}_{4}\)\(e_{4} - e_{5}\)\(4\)
\(\alpha^{(1)}_{5}\)\(e_{5}\)\(4\)
\(\alpha^{(1)}_{6}\)\(e_{2} - e_{4}\)\(4\)
\(\alpha^{(1)}_{7}\)\(e_{3}\)\(4\)
\(\alpha^{(1)}_{8}\)\(e_{1}\)\(4\)
\(\alpha^{(2)}_{1}\)\(e_{1} + 2e_{2} + e_{3} - 2e_{4} + 2e_{5} - 2e_{9}\)\(8\)

Notes

The Gram tensor is the section GRAM of the entry LAMBDA9 of the Catalogue of Lattices.

The catalogue entry gives this reference text: SPLAG p. 157.

References

  1. G. Nebe and N. J. A. Sloane, Catalogue of Lattices, entry LAMBDA9. https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/LAMBDA9.html