Tag 0231
\(\Lambda_{18}\)
Lambda18·LAMBDA18
Invariants
| Rank | 18 |
|---|---|
| Signature \((n_+, n_-)\) | (18, 0) |
| Determinant | \(192\) |
Gram tensor
The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{18}\).
| \(e_{1}\) | \(e_{2}\) | \(e_{3}\) | \(e_{4}\) | \(e_{5}\) | \(e_{6}\) | \(e_{7}\) | \(e_{8}\) | \(e_{9}\) | \(e_{10}\) | \(e_{11}\) | \(e_{12}\) | \(e_{13}\) | \(e_{14}\) | \(e_{15}\) | \(e_{16}\) | \(e_{17}\) | \(e_{18}\) | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(e_{1}\) | 4 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 0 | 0 | 0 | 1 |
| \(e_{2}\) | 2 | 4 | -2 | -2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 |
| \(e_{3}\) | 0 | -2 | 4 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 0 | 0 | 1 | 0 |
| \(e_{4}\) | 0 | -2 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 0 | 0 | 0 | 1 |
| \(e_{5}\) | 0 | 0 | -2 | 0 | 4 | 0 | 0 | 0 | 1 | -1 | 0 | 0 | 1 | 0 | 0 | 0 | -2 | 0 |
| \(e_{6}\) | -2 | -2 | 0 | 0 | 0 | 4 | -2 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(e_{7}\) | 0 | 0 | 0 | 0 | 0 | -2 | 4 | -2 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | -1 |
| \(e_{8}\) | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 4 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 |
| \(e_{9}\) | 0 | 0 | 0 | 0 | 1 | -1 | 0 | 0 | 4 | 0 | -2 | 0 | 1 | 1 | 0 | -1 | 0 | 1 |
| \(e_{10}\) | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 1 | 1 | -1 | 1 | 0 | 0 |
| \(e_{11}\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 4 | -2 | 0 | -1 | 0 | 0 | 0 | -1 |
| \(e_{12}\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 4 | -1 | 1 | 0 | 0 | -1 | 1 |
| \(e_{13}\) | 1 | 0 | -1 | 1 | 1 | 0 | 0 | -1 | 1 | 1 | 0 | -1 | 4 | 0 | 0 | 1 | 0 | 1 |
| \(e_{14}\) | -1 | -1 | 1 | -1 | 0 | 0 | 1 | 0 | 1 | 1 | -1 | 1 | 0 | 4 | 1 | 1 | 0 | 0 |
| \(e_{15}\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 1 | 4 | 0 | 0 | 0 |
| \(e_{16}\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 0 | 0 | 1 | 1 | 0 | 4 | 0 | 0 |
| \(e_{17}\) | 0 | 0 | 1 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 4 | 1 |
| \(e_{18}\) | 1 | 0 | 0 | 1 | 0 | 0 | -1 | 0 | 1 | 0 | -1 | 1 | 1 | 0 | 0 | 0 | 1 | 4 |
[[4, 2, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 1], [2, 4, -2, -2, 0, -2, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0], [0, -2, 4, 0, -2, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 1, 0], [0, -2, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 1], [0, 0, -2, 0, 4, 0, 0, 0, 1, -1, 0, 0, 1, 0, 0, 0, -2, 0], [-2, -2, 0, 0, 0, 4, -2, 0, -1, 0, 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, -1], [0, 0, 0, 0, 0, 0, -2, 4, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, -1, 0, 0, 4, 0, -2, 0, 1, 1, 0, -1, 0, 1], [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, -2, 0, 4, -2, 0, -1, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 4, -1, 1, 0, 0, -1, 1], [1, 0, -1, 1, 1, 0, 0, -1, 1, 1, 0, -1, 4, 0, 0, 1, 0, 1], [-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, 0, -1, 0, 0, 0, 1, 4, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 1, 1, 0, 4, 0, 0], [0, 0, 1, 0, -2, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 4, 1], [1, 0, 0, 1, 0, 0, -1, 0, 1, 0, -1, 1, 1, 0, 0, 0, 1, 4]]
Integral lattice
| Parity | even |
|---|---|
| Discriminant group | \((\mathbb{Z}/2)^{5} \oplus \mathbb{Z}/6\) |
| Unimodular | no |
| Integral overlattices \(L \subseteq M \subseteq L^*\) | 514 |
| Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\) | \(\{2, 3\}\) |
| Character of the discriminant \(\chi_D\), \(D = (-1)^{9} \det L\) | \(\chi_{-3} = \left(\frac{-3}{\cdot}\right)\) |
| Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\) | \(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 17)\, L^\Sigma(s - 9, \chi_{-3})}{L^\Sigma(s - 8, \chi_{-3})}\) |
| Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\) | \(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 17)}{L^\Sigma(s - 8, \chi_{-3})}\) |
| Genus symbol | \(\mathrm{II}_{18,0}\ \bigl(2\colon 1^{12}:2^{-6};\ 3\colon 1^{17}\,3^{1}\bigr)\) |
Definite lattice
| Minimum of \(b(x, x)\), \(x \neq 0\) | \(4\) |
|---|---|
| Kissing number | 7398 |
| Order of the isometry group | \(159252480 = 2^{17} \cdot 3^{5} \cdot 5\) |
Theta series
\[ \sum_{x \in L} q^{b(x, x)} = 1 + 7398q^{4} + O(q^{5}) \]Roots
| Root lattice: \(L = \mathbb{Z}\Phi(L)\) | no |
|---|---|
| Number of roots | 0 |
| \(R(L) = \mathbb{Z}\Phi(L)\) | \(0\) |
| 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
LAMBDA18 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
- G. Nebe and N. J. A. Sloane, Catalogue of Lattices, entry LAMBDA18. https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/LAMBDA18.html