Tag 02AX

\(\Lambda_{\mathfrak{G}_{70}}(-1)\)

Lambda_G(-1), G = G_70 of Hashimoto·coinvariant lattice of the symplectic K3 group G_70 of Hashimoto, twisted by -1

Invariants

Rank19
Signature \((n_+, n_-)\)(19, 0)
Determinant\(300\)

Gram tensor

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

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)\(e_{13}\)\(e_{14}\)\(e_{15}\)\(e_{16}\)\(e_{17}\)\(e_{18}\)\(e_{19}\)
\(e_{1}\)42-2212-2-222-111-1-20-12-1
\(e_{2}\)24-2022000112-1-2-1011-2
\(e_{3}\)-2-24-20-120-2-2-1-1-1210001
\(e_{4}\)20-24-10-2-221-1-1100-1-21-1
\(e_{5}\)120-14010-1112-1-2-2-122-1
\(e_{6}\)22-10040000-101-10001-1
\(e_{7}\)-202-21042-2-110-2010200
\(e_{8}\)-200-20024-1-11100012-10
\(e_{9}\)20-22-10-2-141-1110-10-201
\(e_{10}\)21-2110-1-114110-2-11110
\(e_{11}\)-11-1-11-111-1141-1-1002-10
\(e_{12}\)12-1-1200111140-1-20110
\(e_{13}\)1-1-11-11-2010-1040-10-110
\(e_{14}\)-1-220-2-1000-2-1-10410-2-11
\(e_{15}\)-2-110-2010-1-10-2-11400-11
\(e_{16}\)000-1-1001010000041-10
\(e_{17}\)-110-22022-2121-1-201400
\(e_{18}\)2101210-101-111-1-1-104-1
\(e_{19}\)-1-21-1-1-100100001100-14
 
[[4, 2, -2, 2, 1, 2, -2, -2, 2, 2, -1, 1, 1, -1, -2, 0, -1, 2, -1], [2, 4, -2, 0, 2, 2, 0, 0, 0, 1, 1, 2, -1, -2, -1, 0, 1, 1, -2], [-2, -2, 4, -2, 0, -1, 2, 0, -2, -2, -1, -1, -1, 2, 1, 0, 0, 0, 1], [2, 0, -2, 4, -1, 0, -2, -2, 2, 1, -1, -1, 1, 0, 0, -1, -2, 1, -1], [1, 2, 0, -1, 4, 0, 1, 0, -1, 1, 1, 2, -1, -2, -2, -1, 2, 2, -1], [2, 2, -1, 0, 0, 4, 0, 0, 0, 0, -1, 0, 1, -1, 0, 0, 0, 1, -1], [-2, 0, 2, -2, 1, 0, 4, 2, -2, -1, 1, 0, -2, 0, 1, 0, 2, 0, 0], [-2, 0, 0, -2, 0, 0, 2, 4, -1, -1, 1, 1, 0, 0, 0, 1, 2, -1, 0], [2, 0, -2, 2, -1, 0, -2, -1, 4, 1, -1, 1, 1, 0, -1, 0, -2, 0, 1], [2, 1, -2, 1, 1, 0, -1, -1, 1, 4, 1, 1, 0, -2, -1, 1, 1, 1, 0], [-1, 1, -1, -1, 1, -1, 1, 1, -1, 1, 4, 1, -1, -1, 0, 0, 2, -1, 0], [1, 2, -1, -1, 2, 0, 0, 1, 1, 1, 1, 4, 0, -1, -2, 0, 1, 1, 0], [1, -1, -1, 1, -1, 1, -2, 0, 1, 0, -1, 0, 4, 0, -1, 0, -1, 1, 0], [-1, -2, 2, 0, -2, -1, 0, 0, 0, -2, -1, -1, 0, 4, 1, 0, -2, -1, 1], [-2, -1, 1, 0, -2, 0, 1, 0, -1, -1, 0, -2, -1, 1, 4, 0, 0, -1, 1], [0, 0, 0, -1, -1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 4, 1, -1, 0], [-1, 1, 0, -2, 2, 0, 2, 2, -2, 1, 2, 1, -1, -2, 0, 1, 4, 0, 0], [2, 1, 0, 1, 2, 1, 0, -1, 0, 1, -1, 1, 1, -1, -1, -1, 0, 4, -1], [-1, -2, 1, -1, -1, -1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, -1, 4]]

Integral lattice

Parityeven
Discriminant group\(\mathbb{Z}/5 \oplus \mathbb{Z}/60\)
Unimodularno
Integral overlattices \(L \subseteq M \subseteq L^*\)2
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)\(\{2, 3, 5\}\)
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 \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)\(\zeta^\Sigma(X_n, s) = \zeta^\Sigma(s - 18)\, L^\Sigma(s - 9, \chi_{-300n})\)

Definite lattice

Minimum of \(b(x, x)\), \(x \neq 0\)\(4\)
Kissing number6942

Theta series

\[ \sum_{x \in L} q^{b(x, x)} = 1 + 6942q^{4} + O(q^{5}) \]

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)no
Number of roots2
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong \langle 4 \rangle\)
Rank of \(R(L)\)1
\([L : R(L)]\)infinite: the rank of \(R(L)\) is less than the rank of \(L\)
\(R(L)\) is primitive in \(L\)yes

Components of \(\Phi(L)\)

ComponentTypeValues of \(b(r, r)\)Number of roots\(\mathbb{Z}\Phi_c\)
\(\Phi_{1}\)\(A_{1}\)\(\{4\}\)2\(\cong \langle 4 \rangle\)

Simple roots

Simple rootCoordinates\(b(r, r)\)
\(\alpha_{1}\)\(2e_{1} - e_{2} - 2e_{4} - e_{5} - e_{6} + e_{9} + e_{15} - e_{16} + e_{17} - 2e_{19}\)\(4\)

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 group \(\mathfrak{G}_{70} \cong \mathfrak{S}_5\) of Table 10.2 of Hashimoto, whose invariant lattice \(\Lambda^G\) is in row 70 of Table 10.3.

The Gram tensor is the LLL reduction of the coinvariant lattice lattices[5,18] of Höhn and Mason: the orthogonal complement in the Leech lattice of the fixed-point lattice \(B\) of rank 5 that is number 18 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 70 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.

  • 02AE \(\begin{pmatrix} 4 & 1 \\ 1 & 4 \end{pmatrix} \oplus \langle 20 \rangle\) A lattice \(M\) with \(M \cong L(-1)^\perp\) in \(\mathrm{II}_{3,19}\), where \(L(-1) = \Lambda_G\) for the group \(\mathfrak{G}_{70}\) of Table 10.2 of Hashimoto.
  • 02AF \(\begin{pmatrix} 4 & 2 & 2 \\ 2 & 6 & 1 \\ 2 & 1 & 16 \end{pmatrix}\) A lattice \(M\) with \(M \cong L(-1)^\perp\) in \(\mathrm{II}_{3,19}\), where \(L(-1) = \Lambda_G\) for the group \(\mathfrak{G}_{70}\) 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 \(\mathfrak{G}_{70}\) 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[5,18].