Tag 02B0

\(\Lambda_{\mathfrak{G}_{54}}(-1)\)

Lambda_G(-1), G = G_54 of Hashimoto·coinvariant lattice of the symplectic K3 group G_54 of Hashimoto, twisted by -1

Invariants

Rank19
Signature \((n_+, n_-)\)(19, 0)
Determinant\(384\)

Gram tensor

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

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)\(e_{13}\)\(e_{14}\)\(e_{15}\)\(e_{16}\)\(e_{17}\)\(e_{18}\)\(e_{19}\)
\(e_{1}\)4-2-2-2-1-2212-111122-1211
\(e_{2}\)-2422-12-2-100-2-2100-1-1-2-2
\(e_{3}\)-2240-110-2-1-1001-1-1-1-2-1-1
\(e_{4}\)-220401-2102-1-2-1001-1-2-2
\(e_{5}\)-1-1-104-10-101110002120
\(e_{6}\)-2211-14-10-11-2-2-1-2-1-1-1-2-1
\(e_{7}\)2-20-20-14-10021100-1010
\(e_{8}\)1-1-21-10-141100-2101000
\(e_{9}\)20-100-1014-111122-1200
\(e_{10}\)-10-121101-14-1-2-20-12-10-2
\(e_{11}\)1-20-11-2201-1421000011
\(e_{12}\)1-20-21-2101-2241000122
\(e_{13}\)111-10-11-21-211411-2100
\(e_{14}\)20-100-2012000142011-1
\(e_{15}\)20-100-1002-100124-1200
\(e_{16}\)-1-1-112-1-11-1200-20-14010
\(e_{17}\)2-1-2-11-1002-1011120411
\(e_{18}\)1-2-1-22-21000120101141
\(e_{19}\)1-2-1-20-1000-2120-100114
 
[[4, -2, -2, -2, -1, -2, 2, 1, 2, -1, 1, 1, 1, 2, 2, -1, 2, 1, 1], [-2, 4, 2, 2, -1, 2, -2, -1, 0, 0, -2, -2, 1, 0, 0, -1, -1, -2, -2], [-2, 2, 4, 0, -1, 1, 0, -2, -1, -1, 0, 0, 1, -1, -1, -1, -2, -1, -1], [-2, 2, 0, 4, 0, 1, -2, 1, 0, 2, -1, -2, -1, 0, 0, 1, -1, -2, -2], [-1, -1, -1, 0, 4, -1, 0, -1, 0, 1, 1, 1, 0, 0, 0, 2, 1, 2, 0], [-2, 2, 1, 1, -1, 4, -1, 0, -1, 1, -2, -2, -1, -2, -1, -1, -1, -2, -1], [2, -2, 0, -2, 0, -1, 4, -1, 0, 0, 2, 1, 1, 0, 0, -1, 0, 1, 0], [1, -1, -2, 1, -1, 0, -1, 4, 1, 1, 0, 0, -2, 1, 0, 1, 0, 0, 0], [2, 0, -1, 0, 0, -1, 0, 1, 4, -1, 1, 1, 1, 2, 2, -1, 2, 0, 0], [-1, 0, -1, 2, 1, 1, 0, 1, -1, 4, -1, -2, -2, 0, -1, 2, -1, 0, -2], [1, -2, 0, -1, 1, -2, 2, 0, 1, -1, 4, 2, 1, 0, 0, 0, 0, 1, 1], [1, -2, 0, -2, 1, -2, 1, 0, 1, -2, 2, 4, 1, 0, 0, 0, 1, 2, 2], [1, 1, 1, -1, 0, -1, 1, -2, 1, -2, 1, 1, 4, 1, 1, -2, 1, 0, 0], [2, 0, -1, 0, 0, -2, 0, 1, 2, 0, 0, 0, 1, 4, 2, 0, 1, 1, -1], [2, 0, -1, 0, 0, -1, 0, 0, 2, -1, 0, 0, 1, 2, 4, -1, 2, 0, 0], [-1, -1, -1, 1, 2, -1, -1, 1, -1, 2, 0, 0, -2, 0, -1, 4, 0, 1, 0], [2, -1, -2, -1, 1, -1, 0, 0, 2, -1, 0, 1, 1, 1, 2, 0, 4, 1, 1], [1, -2, -1, -2, 2, -2, 1, 0, 0, 0, 1, 2, 0, 1, 0, 1, 1, 4, 1], [1, -2, -1, -2, 0, -1, 0, 0, 0, -2, 1, 2, 0, -1, 0, 0, 1, 1, 4]]

Integral lattice

Parityeven
Discriminant group\(\mathbb{Z}/2 \oplus \mathbb{Z}/8 \oplus \mathbb{Z}/24\)
Unimodularno
Integral overlattices \(L \subseteq M \subseteq L^*\)14
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)\(\{2, 3\}\)
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 \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)\(\zeta^\Sigma(X_n, s) = \zeta^\Sigma(s - 18)\, L^\Sigma(s - 9, \chi_{-384n})\)

Definite lattice

Minimum of \(b(x, x)\), \(x \neq 0\)\(4\)
Kissing number6042

Theta series

\[ \sum_{x \in L} q^{b(x, x)} = 1 + 6042q^{4} + O(q^{5}) \]

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

Let \(\Lambda = \mathrm{II}_{3,19}\) be the K3 lattice and \(G \subset O(\Lambda)\) a finite group of symplectic isometries. The coinvariant lattice \(\Lambda_G = (\Lambda^G)^\perp\) is negative definite of rank 19 and has no vector of norm \(-2\); this record is \(\Lambda_G(-1)\) for the groups \(\mathfrak{G}_{54} \cong T_{48}\), \(\mathfrak{G}_{38} \cong T_{24}\) of Table 10.2 of Hashimoto, whose invariant lattice \(\Lambda^G\) is in row 54 of Table 10.3.

The Gram tensor is the LLL reduction of the coinvariant lattice lattices[5,23] of Höhn and Mason: the orthogonal complement in the Leech lattice of the fixed-point lattice \(B\) of rank 5 that is number 23 of rank 5 in their Table 1. It satisfies their K3 criterion \(\operatorname{rk} B \geq 5\), \(\operatorname{rk} B - \operatorname{rk} A_B \geq 2\), and among the coinvariant lattices that satisfy it, it is the only one whose rank and discriminant form are those of \(\Lambda_G(-1)\): its discriminant form is the form \(q\) of row 54 of Table 10.3. The sum of its twist by \(-1\) and \(\Lambda^G\) therefore glues to an even unimodular lattice of signature \((3, 19)\), on which the stabilizer of \(B\) in \(\mathrm{Co}_0\) acts with this coinvariant lattice.

  • 02A7 \(\langle 1 \rangle \oplus A_{2}(4)\) A lattice \(M\) with \(M(2) \cong L(-1)^\perp\) in \(\mathrm{II}_{3,19}\), where \(L(-1) = \Lambda_G\) for the group \(\mathfrak{G}_{54}\) 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 \(\mathfrak{G}_{54}\) 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. K. Hashimoto, "Finite symplectic actions on the K3 lattice", arXiv:1012.2682, Tables 10.2 and 10.3.
  2. 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[5,23].