Tag 0166
\(\Lambda_{13}\)
Lambda13·LAMBDA13
Invariants
| Rank | 13 |
|---|---|
| Signature \((n_+, n_-)\) | (13, 0) |
| Determinant | \(1024\) |
Gram tensor
The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{13}\).
| \(e_{1}\) | \(e_{2}\) | \(e_{3}\) | \(e_{4}\) | \(e_{5}\) | \(e_{6}\) | \(e_{7}\) | \(e_{8}\) | \(e_{9}\) | \(e_{10}\) | \(e_{11}\) | \(e_{12}\) | \(e_{13}\) | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(e_{1}\) | 4 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(e_{2}\) | -2 | 4 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
| \(e_{3}\) | 0 | -2 | 4 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 |
| \(e_{4}\) | 0 | 2 | 0 | 4 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(e_{5}\) | 0 | 0 | 0 | 2 | 4 | 2 | 0 | 0 | 2 | 1 | 0 | 0 | 0 |
| \(e_{6}\) | 0 | 0 | 2 | 2 | 2 | 4 | 2 | 2 | 1 | 2 | 0 | 0 | 1 |
| \(e_{7}\) | 0 | 0 | 0 | 0 | 0 | 2 | 4 | 2 | 0 | 2 | 0 | 0 | 0 |
| \(e_{8}\) | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 4 | 0 | 2 | 0 | 0 | 1 |
| \(e_{9}\) | 0 | 0 | 0 | 0 | 2 | 1 | 0 | 0 | 4 | 2 | 0 | 0 | 0 |
| \(e_{10}\) | 0 | 0 | 0 | 0 | 1 | 2 | 2 | 2 | 2 | 4 | 2 | 2 | 1 |
| \(e_{11}\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 4 | 2 | 2 |
| \(e_{12}\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 4 | 1 |
| \(e_{13}\) | 0 | -1 | 2 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 2 | 1 | 4 |
[[4, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 4, -2, 2, 0, 0, 0, 0, 0, 0, 0, 0, -1], [0, -2, 4, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2], [0, 2, 0, 4, 2, 2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 2, 4, 2, 0, 0, 2, 1, 0, 0, 0], [0, 0, 2, 2, 2, 4, 2, 2, 1, 2, 0, 0, 1], [0, 0, 0, 0, 0, 2, 4, 2, 0, 2, 0, 0, 0], [0, 0, 0, 0, 0, 2, 2, 4, 0, 2, 0, 0, 1], [0, 0, 0, 0, 2, 1, 0, 0, 4, 2, 0, 0, 0], [0, 0, 0, 0, 1, 2, 2, 2, 2, 4, 2, 2, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 2, 2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 4, 1], [0, -1, 2, 0, 0, 1, 0, 1, 0, 1, 2, 1, 4]]
Integral lattice
| Parity | even |
|---|---|
| Discriminant group | \((\mathbb{Z}/2)^{4} \oplus (\mathbb{Z}/4)^{3}\) |
| Unimodular | no |
| 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 \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\) | \(\zeta^\Sigma(X_0, s) = \zeta^\Sigma(s - 12)\) |
| Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\) | \(\zeta^\Sigma(X_n, s) = \zeta^\Sigma(s - 12)\, L^\Sigma(s - 6, \chi_{1024n})\) |
| Genus symbol | \(\mathrm{II}_{13,0}\ \bigl(2\colon 1^{6}:2^{4}\,[4^{3}]_{5}\bigr)\) |
Definite lattice
| Minimum of \(b(x, x)\), \(x \neq 0\) | \(4\) |
|---|---|
| Kissing number | 906 |
| Order of the isometry group | \(4718592 = 2^{19} \cdot 3^{2}\) |
Theta series
\[ \sum_{x \in L} q^{b(x, x)} = 1 + 906q^{4} + O(q^{5}) \]Roots
| Root lattice: \(L = \mathbb{Z}\Phi(L)\) | no |
|---|---|
| Number of roots | 26 |
| \(R(L) = \mathbb{Z}\Phi(L)\) | \(\cong \langle 4 \rangle^{13}\) |
| Rank of \(R(L)\) | 13 |
| \([L : R(L)]\) | 256 |
| \(R(L)\) is primitive in \(L\) | no: \(L / R(L) \cong (\mathbb{Z}/2)^{8}\) |
Components of \(\Phi(L)\)
| Component | Type | Values of \(b(r, r)\) | Number of roots | \(\mathbb{Z}\Phi_c\) |
|---|---|---|---|---|
| \(\Phi_{1}\) | \(A_{1}\) | \(\{4\}\) | 2 | \(\cong \langle 4 \rangle\) |
| \(\Phi_{2}\) | \(A_{1}\) | \(\{4\}\) | 2 | \(\cong \langle 4 \rangle\) |
| \(\Phi_{3}\) | \(A_{1}\) | \(\{4\}\) | 2 | \(\cong \langle 4 \rangle\) |
| \(\Phi_{4}\) | \(A_{1}\) | \(\{4\}\) | 2 | \(\cong \langle 4 \rangle\) |
| \(\Phi_{5}\) | \(A_{1}\) | \(\{4\}\) | 2 | \(\cong \langle 4 \rangle\) |
| \(\Phi_{6}\) | \(A_{1}\) | \(\{4\}\) | 2 | \(\cong \langle 4 \rangle\) |
| \(\Phi_{7}\) | \(A_{1}\) | \(\{4\}\) | 2 | \(\cong \langle 4 \rangle\) |
| \(\Phi_{8}\) | \(A_{1}\) | \(\{4\}\) | 2 | \(\cong \langle 4 \rangle\) |
| \(\Phi_{9}\) | \(A_{1}\) | \(\{4\}\) | 2 | \(\cong \langle 4 \rangle\) |
| \(\Phi_{10}\) | \(A_{1}\) | \(\{4\}\) | 2 | \(\cong \langle 4 \rangle\) |
| \(\Phi_{11}\) | \(A_{1}\) | \(\{4\}\) | 2 | \(\cong \langle 4 \rangle\) |
| \(\Phi_{12}\) | \(A_{1}\) | \(\{4\}\) | 2 | \(\cong \langle 4 \rangle\) |
| \(\Phi_{13}\) | \(A_{1}\) | \(\{4\}\) | 2 | \(\cong \langle 4 \rangle\) |
Simple roots
\(\alpha^{(c)}_j\) is simple root \(j\) of the component \(\Phi_c\).
| Simple root | Coordinates | \(b(r, r)\) |
|---|---|---|
| \(\alpha^{(1)}_{1}\) | \(e_{11}\) | \(4\) |
| \(\alpha^{(2)}_{1}\) | \(e_{7}\) | \(4\) |
| \(\alpha^{(3)}_{1}\) | \(e_{4}\) | \(4\) |
| \(\alpha^{(4)}_{1}\) | \(e_{3}\) | \(4\) |
| \(\alpha^{(5)}_{1}\) | \(e_{3} + e_{4} - 2e_{6} + 2e_{10} - e_{11}\) | \(4\) |
| \(\alpha^{(6)}_{1}\) | \(e_{3} + e_{4} - 2e_{6} + e_{7}\) | \(4\) |
| \(\alpha^{(7)}_{1}\) | \(e_{1}\) | \(4\) |
| \(\alpha^{(8)}_{1}\) | \(e_{1} + 2e_{2} + e_{3} - e_{4}\) | \(4\) |
| \(\alpha^{(9)}_{1}\) | \(e_{1} + 2e_{2} + 2e_{3} - e_{4} + 2e_{5} - 2e_{6} - 2e_{9} + 2e_{10} - e_{11}\) | \(4\) |
| \(\alpha^{(10)}_{1}\) | \(e_{1} + 2e_{2} + 2e_{3} - e_{4} + 2e_{5} - 2e_{6} + e_{7}\) | \(4\) |
| \(\alpha^{(11)}_{1}\) | \(e_{1} + 2e_{2} + 3e_{3} + 2e_{5} - 4e_{6} - 2e_{9} + 4e_{10} - e_{11} - 2e_{12}\) | \(4\) |
| \(\alpha^{(12)}_{1}\) | \(e_{1} + 2e_{2} + 3e_{3} + 2e_{5} - 4e_{6} + e_{7} + 2e_{8}\) | \(4\) |
| \(\alpha^{(13)}_{1}\) | \(e_{1} + 2e_{2} + 4e_{3} + 2e_{5} - 4e_{6} + e_{7} + 2e_{8} + e_{11} - 2e_{13}\) | \(4\) |
Notes
The Gram tensor is the section GRAM of the entry
LAMBDA13 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 LAMBDA13. https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/LAMBDA13.html