Tag 0245

\(\Lambda_{19}\)

Lambda19·LAMBDA19

Invariants

Rank19
Signature \((n_+, n_-)\)(19, 0)
Determinant\(128\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{19}\).

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)\(e_{13}\)\(e_{14}\)\(e_{15}\)\(e_{16}\)\(e_{17}\)\(e_{18}\)\(e_{19}\)
\(e_{1}\)42000-20000001-100010
\(e_{2}\)24-2-20-20000000-100000
\(e_{3}\)0-240-20000000-1100100
\(e_{4}\)0-204000000001-10001-1
\(e_{5}\)00-2040001-1001000-200
\(e_{6}\)-2-20004-20-10000000001
\(e_{7}\)00000-24-2000001000-10
\(e_{8}\)000000-240000-100000-1
\(e_{9}\)00001-10040-20110-1010
\(e_{10}\)0000-1000040011-11000
\(e_{11}\)00000000-204-20-1000-10
\(e_{12}\)0000000000-24-1100-110
\(e_{13}\)10-11100-1110-14001011
\(e_{14}\)-1-11-1001011-110411001
\(e_{15}\)000000000-1000140001
\(e_{16}\)00000000-11001104000
\(e_{17}\)0010-2000000-10000411
\(e_{18}\)100100-1010-111000140
\(e_{19}\)000-1010-100001110104
 
[[4, 2, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 1, 0], [2, 4, -2, -2, 0, -2, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [0, -2, 4, 0, -2, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 1, 0, 0], [0, -2, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 1, -1], [0, 0, -2, 0, 4, 0, 0, 0, 1, -1, 0, 0, 1, 0, 0, 0, -2, 0, 0], [-2, -2, 0, 0, 0, 4, -2, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, -2, 4, -2, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, 0, -2, 4, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 1, -1, 0, 0, 4, 0, -2, 0, 1, 1, 0, -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, 0, 0, -2, 0, 4, -2, 0, -1, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 4, -1, 1, 0, 0, -1, 1, 0], [1, 0, -1, 1, 1, 0, 0, -1, 1, 1, 0, -1, 4, 0, 0, 1, 0, 1, 1], [-1, -1, 1, -1, 0, 0, 1, 0, 1, 1, -1, 1, 0, 4, 1, 1, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 1, 4, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 1, 1, 0, 4, 0, 0, 0], [0, 0, 1, 0, -2, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 4, 1, 1], [1, 0, 0, 1, 0, 0, -1, 0, 1, 0, -1, 1, 1, 0, 0, 0, 1, 4, 0], [0, 0, 0, -1, 0, 1, 0, -1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 4]]

Integral lattice

Parityeven
Discriminant group\((\mathbb{Z}/2)^{4} \oplus \mathbb{Z}/8\)
Unimodularno
Integral overlattices \(L \subseteq M \subseteq L^*\)122
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)\(\{2\}\)
Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)\(\zeta^\Sigma(X_0, s) = \zeta^\Sigma(s - 18)\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)\(\zeta^\Sigma(X_n, s) = \zeta^\Sigma(s - 18)\, L^\Sigma(s - 9, \chi_{-128n})\)
Genus symbol\(\mathrm{II}_{19,0}\ \bigl(2\colon 1^{14}:2^{-4}:[8^{-1}]_{3}\bigr)\)

Definite lattice

Minimum of \(b(x, x)\), \(x \neq 0\)\(4\)
Kissing number10668
Order of the isometry group\(23592960 = 2^{19} \cdot 3^{2} \cdot 5\)

Theta series

\[ \sum_{x \in L} q^{b(x, x)} = 1 + 10668q^{4} + O(q^{5}) \]

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)no
Number of roots0
\(R(L) = \mathbb{Z}\Phi(L)\)\(0\)
Rank of \(R(L)\)0
\([L : R(L)]\)infinite: the rank of \(R(L)\) is less than the rank of \(L\)
\(R(L)\) is primitive in \(L\)yes

Notes

The Gram tensor is the section GRAM of the entry LAMBDA19 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 LAMBDA19. https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/LAMBDA19.html