Tag 0149

\(\Lambda_{12}\)

Lambda12·LAMBDA12

Invariants

Rank12
Signature \((n_+, n_-)\)(12, 0)
Determinant\(1024\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{12}\).

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)
\(e_{1}\)4-20000000000
\(e_{2}\)-24-2200000000
\(e_{3}\)0-24002000000
\(e_{4}\)020422000000
\(e_{5}\)000242002100
\(e_{6}\)002224221200
\(e_{7}\)000002420200
\(e_{8}\)000002240200
\(e_{9}\)000021004200
\(e_{10}\)000012222422
\(e_{11}\)000000000242
\(e_{12}\)000000000224
 
[[4, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 4, -2, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, -2, 4, 0, 0, 2, 0, 0, 0, 0, 0, 0], [0, 2, 0, 4, 2, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 2, 4, 2, 0, 0, 2, 1, 0, 0], [0, 0, 2, 2, 2, 4, 2, 2, 1, 2, 0, 0], [0, 0, 0, 0, 0, 2, 4, 2, 0, 2, 0, 0], [0, 0, 0, 0, 0, 2, 2, 4, 0, 2, 0, 0], [0, 0, 0, 0, 2, 1, 0, 0, 4, 2, 0, 0], [0, 0, 0, 0, 1, 2, 2, 2, 2, 4, 2, 2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 4]]

Integral lattice

Parityeven
Discriminant group\((\mathbb{Z}/2)^{6} \oplus (\mathbb{Z}/4)^{2}\)
Unimodularno
Integral overlattices \(L \subseteq M \subseteq L^*\)not decided
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)\(\{2\}\)
Character of the discriminant \(\chi_D\), \(D = (-1)^{6} \det L\)trivial
Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)\(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 11)\, \zeta^\Sigma(s - 6)}{\zeta^\Sigma(s - 5)}\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)\(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 11)}{\zeta^\Sigma(s - 5)}\)
Genus symbol\(\mathrm{II}_{12,0}\ \bigl(2\colon 1^{4}:2^{-6}:4^{-2}\bigr)\)
Class number of the genus7

Definite lattice

Minimum of \(b(x, x)\), \(x \neq 0\)\(4\)
Kissing number648
Order of the isometry group\(254803968 = 2^{20} \cdot 3^{5}\)

Theta series

\[ \sum_{x \in L} q^{b(x, x)} = 1 + 648q^{4} + 3072q^{6} + O(q^{7}) \]

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)no
Number of roots72
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong D_{4}(2)^{3}\)
Rank of \(R(L)\)12
\([L : R(L)]\)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\(\cong D_{4}(2)\)
\(\Phi_{3}\)\(D_{4}\)\(\{4\}\)24\(\cong D_{4}(2)\)

Simple roots

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

Simple rootCoordinates\(b(r, r)\)
\(\alpha^{(1)}_{1}\)\(e_{4}\)\(4\)
\(\alpha^{(1)}_{2}\)\(e_{2} - e_{4}\)\(4\)
\(\alpha^{(1)}_{3}\)\(e_{3}\)\(4\)
\(\alpha^{(1)}_{4}\)\(e_{1}\)\(4\)
\(\alpha^{(2)}_{1}\)\(e_{3} + e_{4} - 2e_{6} + e_{8}\)\(4\)
\(\alpha^{(2)}_{2}\)\(e_{7} - e_{8}\)\(4\)
\(\alpha^{(2)}_{3}\)\(e_{1} + 2e_{2} + 2e_{3} - e_{4} + 2e_{5} - 2e_{6} + e_{8}\)\(4\)
\(\alpha^{(2)}_{4}\)\(e_{8}\)\(4\)
\(\alpha^{(3)}_{1}\)\(e_{1} + 2e_{2} + 2e_{3} - e_{4} + 2e_{5} - 2e_{6} - 2e_{9} + 2e_{10} - e_{11} - e_{12}\)\(4\)
\(\alpha^{(3)}_{2}\)\(e_{12}\)\(4\)
\(\alpha^{(3)}_{3}\)\(e_{11} - e_{12}\)\(4\)
\(\alpha^{(3)}_{4}\)\(e_{3} + e_{4} - 2e_{6} + 2e_{10} - e_{11} - e_{12}\)\(4\)

Notes

The Gram tensor is the section GRAM of the entry LAMBDA12 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

Let \(\Lambda = \mathrm{II}_{3,19}\) be the K3 lattice. For the group \(\mathfrak{G}_{3} \cong C_2^2\) of symplectic isometries in Table 10.2 of Hashimoto, the coinvariant lattice \(\Lambda_G = (\Lambda^G)^\perp\) twisted by \(-1\) is this lattice; \(\Lambda^G\) is in row 3 of Table 10.3. The coinvariant lattice lattices[12,1] of Höhn and Mason, the orthogonal complement in the Leech lattice of their fixed-point lattice number 1 of rank 12, is isometric to this lattice, and its discriminant form is the form \(q\) of row 3 of Table 10.3.

  • 029N \(U \oplus U(2)^{2} \oplus D_{4}(-2)\) A lattice \(M\) with \(M \cong L(-1)^\perp\) in \(\mathrm{II}_{3,19}\), where \(L(-1) = \Lambda_G\) for the group \(\mathfrak{G}_{3}\) of Table 10.2 of Hashimoto.
  • 027E \(U^{3} \oplus E_{8}(-1)^{2}\) The K3 lattice \(\mathrm{II}_{3,19}\), in which \(L(-1)\) is the coinvariant lattice \(\Lambda_G\) of the group \(\mathfrak{G}_{3}\) of Table 10.2 of Hashimoto.
  • 028S \(\Lambda_{24}\) The Leech lattice \(\Lambda\), in which \(L\) is the orthogonal complement of the sublattice fixed by a subgroup of \(\mathrm{Co}_0\).

Morphisms

References

  1. G. Nebe and N. J. A. Sloane, Catalogue of Lattices, entry LAMBDA12. https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/LAMBDA12.html
  2. K. Hashimoto, "Finite symplectic actions on the K3 lattice", arXiv:1012.2682, Tables 10.2 and 10.3.
  3. G. Höhn and G. Mason, "The 290 fixed-point sublattices of the Leech lattice", arXiv:1505.06420, Table 1 and its ancillary Magma file lattices.txt, entry lattices[12,1].