Tag 029L

\(U^{3} \oplus E_{8}(-2)\)

U^3 + E8(-2)·invariant lattice of the symplectic K3 group G_1 of Hashimoto

Invariants

Rank14
Signature \((n_+, n_-)\)(3, 11)
Determinant\(-256\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{14}\).

In this basis, \(L = {}\)\(\mathbb{Z}\{e_{1}, e_{2}\}\) \(\oplus\) \(\mathbb{Z}\{e_{3}, e_{4}\}\) \(\oplus\) \(\mathbb{Z}\{e_{5}, e_{6}\}\) \(\oplus\) \(\mathbb{Z}\{e_{7}, \dots, e_{14}\}\), an orthogonal sum.

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)\(e_{13}\)\(e_{14}\)
\(e_{1}\)01000000000000
\(e_{2}\)10000000000000
\(e_{3}\)00010000000000
\(e_{4}\)00100000000000
\(e_{5}\)00000100000000
\(e_{6}\)00001000000000
\(e_{7}\)000000-40200000
\(e_{8}\)0000000-4020000
\(e_{9}\)00000020-420000
\(e_{10}\)000000022-42000
\(e_{11}\)0000000002-4200
\(e_{12}\)00000000002-420
\(e_{13}\)000000000002-42
\(e_{14}\)0000000000002-4
 
[[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -4, 0, 2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -4, 0, 2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2, 0, -4, 2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 2, 2, -4, 2, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 2, -4, 2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, -4, 2, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, -4, 2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, -4]]

Integral lattice

Parityeven
Discriminant group\((\mathbb{Z}/2)^{8}\)
Unimodularno
Integral overlattices \(L \subseteq M \subseteq L^*\)not decided
Nikulin invariants \((r, a, \delta)\)\((14, 8, 0)\)
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)\(\{2\}\)
Character of the discriminant \(\chi_D\), \(D = (-1)^{7} \det L\)trivial
Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)\(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 13)\, \zeta^\Sigma(s - 7)}{\zeta^\Sigma(s - 6)}\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)\(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 13)}{\zeta^\Sigma(s - 6)}\)
Hyperbolic index \(\max\{n : L \cong U^n \oplus L'\}\)\(\geq 3\), by an embedding of \(U^{3}\) in the morphism files

Indefinite lattice

Isotropicyes

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)yes
\(L = \mathbb{Z}\Phi_S(L)\) for\(S = \{-4, 2\}\)
\(R(L) = \mathbb{Z}\Phi(L)\)\(L\)
Rank of \(R(L)\)14
\([L : R(L)]\)1
\(R(L)\) is primitive in \(L\)yes
ProofEach \(r_i\) in roots that generate \(\mathbb{Z}\Phi(L)\) is a root, and the \(r_i\) generate \(L\).

Roots that generate \(\mathbb{Z}\Phi(L)\)

The \(r_i\) are a basis of \(L\).

RootCoordinates\(b(r, r)\)
\(r_{1}\)\(e_{7}\)\(-4\)
\(r_{2}\)\(e_{8}\)\(-4\)
\(r_{3}\)\(e_{9}\)\(-4\)
\(r_{4}\)\(e_{10}\)\(-4\)
\(r_{5}\)\(e_{11}\)\(-4\)
\(r_{6}\)\(e_{12}\)\(-4\)
\(r_{7}\)\(e_{13}\)\(-4\)
\(r_{8}\)\(e_{14}\)\(-4\)
\(r_{9}\)\(e_{1} + e_{2}\)\(2\)
\(r_{10}\)\(e_{3} + e_{4}\)\(2\)
\(r_{11}\)\(e_{5} + e_{6}\)\(2\)
\(r_{12}\)\(e_{1} + e_{2} + e_{3}\)\(2\)
\(r_{13}\)\(e_{1} + e_{2} + e_{5}\)\(2\)
\(r_{14}\)\(e_{1} + e_{3} + e_{4}\)\(2\)

Notes

Let \(\Lambda = \mathrm{II}_{3,19}\) be the K3 lattice and \(G \subset O(\Lambda)\) a finite group of symplectic isometries. Row 1 of Table 10.3 of Hashimoto gives the invariant lattice \(\Lambda^G\) for the group \(\mathfrak{G}_{1} \cong C_2\) of Table 10.2, and this record is \(\Lambda^G\).

  • 0094 \(E_{8}\) A lattice \(M\) with \(M(-2) \cong L^\perp\) in \(\mathrm{II}_{3,19}\), where \(L = \Lambda^G\) for the group \(\mathfrak{G}_{1}\) of Table 10.2 of Hashimoto.
  • 027E \(U^{3} \oplus E_{8}(-1)^{2}\) The K3 lattice \(\mathrm{II}_{3,19}\), in which \(L\) is the invariant lattice \(\Lambda^G\) of the group \(\mathfrak{G}_{1}\) of Table 10.2 of Hashimoto.

Morphisms

References

  1. K. Hashimoto, "Finite symplectic actions on the K3 lattice", arXiv:1012.2682, Tables 10.2 and 10.3.