Tag 0123

\(\Lambda_{10}\)

Lambda10·LAMBDA10

Invariants

Rank10
Signature \((n_+, n_-)\)(10, 0)
Determinant\(768\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{10}\).

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)
\(e_{1}\)4-200000000
\(e_{2}\)-24-22000000
\(e_{3}\)0-240020000
\(e_{4}\)0204220000
\(e_{5}\)0002420021
\(e_{6}\)0022242212
\(e_{7}\)0000024202
\(e_{8}\)0000022402
\(e_{9}\)0000210042
\(e_{10}\)0000122224
 
[[4, -2, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 4, -2, 2, 0, 0, 0, 0, 0, 0], [0, -2, 4, 0, 0, 2, 0, 0, 0, 0], [0, 2, 0, 4, 2, 2, 0, 0, 0, 0], [0, 0, 0, 2, 4, 2, 0, 0, 2, 1], [0, 0, 2, 2, 2, 4, 2, 2, 1, 2], [0, 0, 0, 0, 0, 2, 4, 2, 0, 2], [0, 0, 0, 0, 0, 2, 2, 4, 0, 2], [0, 0, 0, 0, 2, 1, 0, 0, 4, 2], [0, 0, 0, 0, 1, 2, 2, 2, 2, 4]]

Integral lattice

Parityeven
Discriminant group\((\mathbb{Z}/2)^{4} \oplus \mathbb{Z}/4 \oplus \mathbb{Z}/12\)
Unimodularno
Integral overlattices \(L \subseteq M \subseteq L^*\)605
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)\(\{2, 3\}\)
Character of the discriminant \(\chi_D\), \(D = (-1)^{5} \det L\)\(\chi_{-3} = \left(\frac{-3}{\cdot}\right)\)
Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)\(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 9)\, L^\Sigma(s - 5, \chi_{-3})}{L^\Sigma(s - 4, \chi_{-3})}\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)\(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 9)}{L^\Sigma(s - 4, \chi_{-3})}\)
Genus symbol\(\mathrm{II}_{10,0}\ \bigl(2\colon 1^{4}:2^{4}:4^{-2};\ 3\colon 1^{-9}\,3^{-1}\bigr)\)
Class number of the genus9

Definite lattice

Minimum of \(b(x, x)\), \(x \neq 0\)\(4\)
Kissing number336
Order of the isometry group\(884736 = 2^{15} \cdot 3^{3}\)

Theta series

\[ \sum_{x \in L} q^{b(x, x)} = 1 + 336q^{4} + 768q^{6} + O(q^{7}) \]

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)no
Number of roots48
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong D_{4}(2)^{2}\)
Rank of \(R(L)\)8
\([L : R(L)]\)infinite: the rank of \(R(L)\) is less than the rank of \(L\)
\(R(L)\) is primitive in \(L\)no: \(M / R(L) \cong (\mathbb{Z}/2)^{2}\) for \(M = L \cap (R(L) \otimes \mathbb{Q})\)

Components of \(\Phi(L)\)

ComponentTypeValues of \(b(r, r)\)Number of roots\(\mathbb{Z}\Phi_c\)
\(\Phi_{1}\)\(D_{4}\)\(\{4\}\)24\(\cong D_{4}(2)\)
\(\Phi_{2}\)\(D_{4}\)\(\{4\}\)24\(\cong D_{4}(2)\)

Simple roots

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

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

Notes

The Gram tensor is the section GRAM of the entry LAMBDA10 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 LAMBDA10. https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/LAMBDA10.html