Tag 0150

\(K_{12}\)

K12·Coxeter-Todd lattice

Invariants

Rank12
Signature \((n_+, n_-)\)(12, 0)
Determinant\(729\)

Gram tensor

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

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)
\(e_{1}\)400-2002-1-1-12-1
\(e_{2}\)0400-202-1-1-1-12
\(e_{3}\)00400-2222-1-1-1
\(e_{4}\)-200400-1-122-1-1
\(e_{5}\)0-20040-12-12-1-1
\(e_{6}\)00-2004-1-1-1222
\(e_{7}\)222-1-1-1400-200
\(e_{8}\)-1-12-12-10400-20
\(e_{9}\)-1-122-1-100400-2
\(e_{10}\)-1-1-1222-200400
\(e_{11}\)2-1-1-1-120-20040
\(e_{12}\)-12-1-1-1200-2004
 
[[4, 0, 0, -2, 0, 0, 2, -1, -1, -1, 2, -1], [0, 4, 0, 0, -2, 0, 2, -1, -1, -1, -1, 2], [0, 0, 4, 0, 0, -2, 2, 2, 2, -1, -1, -1], [-2, 0, 0, 4, 0, 0, -1, -1, 2, 2, -1, -1], [0, -2, 0, 0, 4, 0, -1, 2, -1, 2, -1, -1], [0, 0, -2, 0, 0, 4, -1, -1, -1, 2, 2, 2], [2, 2, 2, -1, -1, -1, 4, 0, 0, -2, 0, 0], [-1, -1, 2, -1, 2, -1, 0, 4, 0, 0, -2, 0], [-1, -1, 2, 2, -1, -1, 0, 0, 4, 0, 0, -2], [-1, -1, -1, 2, 2, 2, -2, 0, 0, 4, 0, 0], [2, -1, -1, -1, -1, 2, 0, -2, 0, 0, 4, 0], [-1, 2, -1, -1, -1, 2, 0, 0, -2, 0, 0, 4]]

Integral lattice

Parityeven
Discriminant group\((\mathbb{Z}/3)^{6}\)
Unimodularno
Integral overlattices \(L \subseteq M \subseteq L^*\)393
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)\(\{2, 3\}\)
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\(\mathrm{II}_{12,0}\ \bigl(2\colon 1^{12};\ 3\colon 1^{6}\,3^{6}\bigr)\)
Class number of the genus10

Definite lattice

Minimum of \(b(x, x)\), \(x \neq 0\)\(4\)
Kissing number756
Order of the isometry group\(78382080 = 2^{10} \cdot 3^{7} \cdot 5 \cdot 7\)

Theta series

\[ \sum_{x \in L} q^{b(x, x)} = 1 + 756q^{4} + 4032q^{6} + O(q^{7}) \]

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 K12 of the Catalogue of Lattices.

The catalogue entry gives this reference text: For the long history of this lattice, and many constructions, see J. H. Conway and N. J. A. Sloane, The Coxeter-Todd Lattice, The Mitchell Group and Related Sphere Packings, Math. Proc. Comb. Phil. Soc. volume 93 1983 pages 421-440. Also SPLAG 127.

Let \(\Lambda = \mathrm{II}_{3,19}\) be the K3 lattice. For the group \(\mathfrak{G}_{2} \cong C_3\) 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 2 of Table 10.3. The coinvariant lattice lattices[12,2] of Höhn and Mason, the orthogonal complement in the Leech lattice of their fixed-point lattice number 2 of rank 12, is isometric to this lattice, and its discriminant form is the form \(q\) of row 2 of Table 10.3.

  • 029M \(U \oplus U(3)^{2} \oplus A_{2}(-1)^{2}\) A lattice \(M\) with \(M \cong L(-1)^\perp\) in \(\mathrm{II}_{3,19}\), where \(L(-1) = \Lambda_G\) for the group \(\mathfrak{G}_{2}\) 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 \(\mathfrak{G}_{2}\) 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 K12. https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/K12.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[12,2].