Tag 0137

\(\Lambda_{11}\)

Lambda11·LAMBDA11

Invariants

Rank11
Signature \((n_+, n_-)\)(11, 0)
Determinant\(1024\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{11}\).

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)
\(e_{1}\)4-2000000000
\(e_{2}\)-24-220000000
\(e_{3}\)0-2400200000
\(e_{4}\)02042200000
\(e_{5}\)00024200210
\(e_{6}\)00222422120
\(e_{7}\)00000242020
\(e_{8}\)00000224020
\(e_{9}\)00002100420
\(e_{10}\)00001222242
\(e_{11}\)00000000024
 
[[4, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 4, -2, 2, 0, 0, 0, 0, 0, 0, 0], [0, -2, 4, 0, 0, 2, 0, 0, 0, 0, 0], [0, 2, 0, 4, 2, 2, 0, 0, 0, 0, 0], [0, 0, 0, 2, 4, 2, 0, 0, 2, 1, 0], [0, 0, 2, 2, 2, 4, 2, 2, 1, 2, 0], [0, 0, 0, 0, 0, 2, 4, 2, 0, 2, 0], [0, 0, 0, 0, 0, 2, 2, 4, 0, 2, 0], [0, 0, 0, 0, 2, 1, 0, 0, 4, 2, 0], [0, 0, 0, 0, 1, 2, 2, 2, 2, 4, 2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4]]

Integral lattice

Parityeven
Discriminant group\((\mathbb{Z}/2)^{4} \oplus (\mathbb{Z}/4)^{3}\)
Unimodularno
Integral overlattices \(L \subseteq M \subseteq L^*\)not decided
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)\(\{2\}\)
Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)\(\zeta^\Sigma(X_0, s) = \zeta^\Sigma(s - 10)\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)\(\zeta^\Sigma(X_n, s) = \zeta^\Sigma(s - 10)\, L^\Sigma(s - 5, \chi_{-1024n})\)
Genus symbol\(\mathrm{II}_{11,0}\ \bigl(2\colon 1^{4}:2^{4}\,[4^{3}]_{3}\bigr)\)

Definite lattice

Minimum of \(b(x, x)\), \(x \neq 0\)\(4\)
Kissing number438
Order of the isometry group\(3538944 = 2^{17} \cdot 3^{3}\)

Theta series

\[ \sum_{x \in L} q^{b(x, x)} = 1 + 438q^{4} + 1536q^{6} + O(q^{7}) \]

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)no
Number of roots54
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong D_{4}(2)^{2} \oplus \langle 4 \rangle^{3}\)
Rank of \(R(L)\)11
\([L : R(L)]\)16
\(R(L)\) is primitive in \(L\)no: \(L / R(L) \cong (\mathbb{Z}/2)^{4}\)

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\(\cong D_{4}(2)\)
\(\Phi_{3}\)\(A_{1}\)\(\{4\}\)2\(\cong \langle 4 \rangle\)
\(\Phi_{4}\)\(A_{1}\)\(\{4\}\)2\(\cong \langle 4 \rangle\)
\(\Phi_{5}\)\(A_{1}\)\(\{4\}\)2\(\cong \langle 4 \rangle\)

Simple roots

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

Simple rootCoordinates\(b(r, r)\)
\(\alpha^{(1)}_{1}\)\(e_{4}\)\(4\)
\(\alpha^{(1)}_{2}\)\(e_{2} - e_{4}\)\(4\)
\(\alpha^{(1)}_{3}\)\(e_{3}\)\(4\)
\(\alpha^{(1)}_{4}\)\(e_{1}\)\(4\)
\(\alpha^{(2)}_{1}\)\(e_{3} + e_{4} - 2e_{6} + e_{8}\)\(4\)
\(\alpha^{(2)}_{2}\)\(e_{7} - e_{8}\)\(4\)
\(\alpha^{(2)}_{3}\)\(e_{1} + 2e_{2} + 2e_{3} - e_{4} + 2e_{5} - 2e_{6} + e_{8}\)\(4\)
\(\alpha^{(2)}_{4}\)\(e_{8}\)\(4\)
\(\alpha^{(3)}_{1}\)\(e_{11}\)\(4\)
\(\alpha^{(4)}_{1}\)\(e_{3} + e_{4} - 2e_{6} + 2e_{10} - e_{11}\)\(4\)
\(\alpha^{(5)}_{1}\)\(e_{1} + 2e_{2} + 2e_{3} - e_{4} + 2e_{5} - 2e_{6} - 2e_{9} + 2e_{10} - e_{11}\)\(4\)

Notes

The Gram tensor is the section GRAM of the entry LAMBDA11 of the Catalogue of Lattices.

The catalogue entry gives this reference text: 1. SPLAG p. 157. 2. W.Plesken, M.Pohst, Constructing integral lattices with prescribed minimum II, Math. Comp. Vol 60, N. 202 (1993) 817-825

References

  1. G. Nebe and N. J. A. Sloane, Catalogue of Lattices, entry LAMBDA11. https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/LAMBDA11.html