Tag 0219

\(\Lambda_{17}\)

Lambda17·LAMBDA17

Invariants

Rank17
Signature \((n_+, n_-)\)(17, 0)
Determinant\(256\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{17}\).

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)\(e_{13}\)\(e_{14}\)\(e_{15}\)\(e_{16}\)\(e_{17}\)
\(e_{1}\)42000-20000001-1000
\(e_{2}\)24-2-20-20000000-1000
\(e_{3}\)0-240-20000000-11001
\(e_{4}\)0-204000000001-1000
\(e_{5}\)00-2040001-1001000-2
\(e_{6}\)-2-20004-20-100000000
\(e_{7}\)00000-24-2000001000
\(e_{8}\)000000-240000-10000
\(e_{9}\)00001-10040-20110-10
\(e_{10}\)0000-1000040011-110
\(e_{11}\)00000000-204-20-1000
\(e_{12}\)0000000000-24-1100-1
\(e_{13}\)10-11100-1110-140010
\(e_{14}\)-1-11-1001011-1104110
\(e_{15}\)000000000-10001400
\(e_{16}\)00000000-110011040
\(e_{17}\)0010-2000000-100004
 
[[4, 2, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0], [2, 4, -2, -2, 0, -2, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0], [0, -2, 4, 0, -2, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 1], [0, -2, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0], [0, 0, -2, 0, 4, 0, 0, 0, 1, -1, 0, 0, 1, 0, 0, 0, -2], [-2, -2, 0, 0, 0, 4, -2, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -2, 4, -2, 0, 0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, 4, 0, 0, 0, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, 1, -1, 0, 0, 4, 0, -2, 0, 1, 1, 0, -1, 0], [0, 0, 0, 0, -1, 0, 0, 0, 0, 4, 0, 0, 1, 1, -1, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 4, -2, 0, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 4, -1, 1, 0, 0, -1], [1, 0, -1, 1, 1, 0, 0, -1, 1, 1, 0, -1, 4, 0, 0, 1, 0], [-1, -1, 1, -1, 0, 0, 1, 0, 1, 1, -1, 1, 0, 4, 1, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 1, 4, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 1, 1, 0, 4, 0], [0, 0, 1, 0, -2, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 4]]

Integral lattice

Parityeven
Discriminant group\((\mathbb{Z}/2)^{6} \oplus \mathbb{Z}/4\)
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 \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)\(\zeta^\Sigma(X_0, s) = \zeta^\Sigma(s - 16)\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)\(\zeta^\Sigma(X_n, s) = \zeta^\Sigma(s - 16)\, L^\Sigma(s - 8, \chi_{256n})\)
Genus symbol\(\mathrm{II}_{17,0}\ \bigl(2\colon 1^{10}:2^{6}\,[4^{1}]_{1}\bigr)\)

Definite lattice

Minimum of \(b(x, x)\), \(x \neq 0\)\(4\)
Kissing number5346
Order of the isometry group\(1486356480 = 2^{19} \cdot 3^{4} \cdot 5 \cdot 7\)

Theta series

\[ \sum_{x \in L} q^{b(x, x)} = 1 + 5346q^{4} + O(q^{5}) \]

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)no
Number of roots2
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong \langle 4 \rangle\)
Rank of \(R(L)\)1
\([L : R(L)]\)infinite: the rank of \(R(L)\) is less than the rank of \(L\)
\(R(L)\) is primitive in \(L\)yes

Components of \(\Phi(L)\)

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

Simple roots

Simple rootCoordinates\(b(r, r)\)
\(\alpha_{1}\)\(64e_{1} - 390e_{2} - 256e_{3} - 194e_{4} - 121e_{5} - 264e_{6} - 176e_{7} - 90e_{8} - 52e_{9} - 28e_{10} - 37e_{11} - 20e_{12} - 8e_{13} - 4e_{14} - 6e_{15} - 3e_{16} - 2e_{17}\)\(4\)

Notes

The Gram tensor is the section GRAM of the entry LAMBDA17 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 LAMBDA17. https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/LAMBDA17.html