Tag 027G

\(U \oplus E_{8}(-1)^{3}\)

U + E8(-1)^3·II_{1,25}

Invariants

Rank26
Signature \((n_+, n_-)\)(1, 25)
Determinant\(-1\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{26}\).

In this basis, \(L = {}\)\(\mathbb{Z}\{e_{1}, e_{2}\}\) \(\oplus\) \(\mathbb{Z}\{e_{3}, \dots, e_{10}\}\) \(\oplus\) \(\mathbb{Z}\{e_{11}, \dots, e_{18}\}\) \(\oplus\) \(\mathbb{Z}\{e_{19}, \dots, e_{26}\}\), an orthogonal sum.

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)\(e_{13}\)\(e_{14}\)\(e_{15}\)\(e_{16}\)\(e_{17}\)\(e_{18}\)\(e_{19}\)\(e_{20}\)\(e_{21}\)\(e_{22}\)\(e_{23}\)\(e_{24}\)\(e_{25}\)\(e_{26}\)
\(e_{1}\)01000000000000000000000000
\(e_{2}\)10000000000000000000000000
\(e_{3}\)00-201000000000000000000000
\(e_{4}\)000-20100000000000000000000
\(e_{5}\)0010-2100000000000000000000
\(e_{6}\)00011-210000000000000000000
\(e_{7}\)000001-21000000000000000000
\(e_{8}\)0000001-2100000000000000000
\(e_{9}\)00000001-210000000000000000
\(e_{10}\)000000001-20000000000000000
\(e_{11}\)0000000000-2010000000000000
\(e_{12}\)00000000000-201000000000000
\(e_{13}\)000000000010-21000000000000
\(e_{14}\)0000000000011-2100000000000
\(e_{15}\)00000000000001-210000000000
\(e_{16}\)000000000000001-21000000000
\(e_{17}\)0000000000000001-2100000000
\(e_{18}\)00000000000000001-200000000
\(e_{19}\)000000000000000000-20100000
\(e_{20}\)0000000000000000000-2010000
\(e_{21}\)00000000000000000010-210000
\(e_{22}\)000000000000000000011-21000
\(e_{23}\)0000000000000000000001-2100
\(e_{24}\)00000000000000000000001-210
\(e_{25}\)000000000000000000000001-21
\(e_{26}\)0000000000000000000000001-2
 
[[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 1, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -2, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, -2, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2]]

Integral lattice

Parityeven
Discriminant group\(0\)
Unimodularyes
Integral overlattices \(L \subseteq M \subseteq L^*\)1
Nikulin invariants \((r, a, \delta)\)\((26, 0, 0)\)
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)\(\{2\}\)
Character of the discriminant \(\chi_D\), \(D = (-1)^{13} \det L\)trivial
Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)\(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 25)\, \zeta^\Sigma(s - 13)}{\zeta^\Sigma(s - 12)}\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)\(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 25)}{\zeta^\Sigma(s - 12)}\)
Hyperbolic index \(\max\{n : L \cong U^n \oplus L'\}\)\(\geq 1\), by an embedding of \(U^{1}\) in the morphism files

Indefinite lattice

Isotropicyes

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)yes
\(L = \mathbb{Z}\Phi_S(L)\) for\(S = \{-2, 2\}\)
\(R(L) = \mathbb{Z}\Phi(L)\)\(L\)
Rank of \(R(L)\)26
\([L : R(L)]\)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\(b(r, r)\)
\(r_{1}\)\(e_{3}\)\(-2\)
\(r_{2}\)\(e_{4}\)\(-2\)
\(r_{3}\)\(e_{5}\)\(-2\)
\(r_{4}\)\(e_{6}\)\(-2\)
\(r_{5}\)\(e_{7}\)\(-2\)
\(r_{6}\)\(e_{8}\)\(-2\)
\(r_{7}\)\(e_{9}\)\(-2\)
\(r_{8}\)\(e_{10}\)\(-2\)
\(r_{9}\)\(e_{11}\)\(-2\)
\(r_{10}\)\(e_{12}\)\(-2\)
\(r_{11}\)\(e_{13}\)\(-2\)
\(r_{12}\)\(e_{14}\)\(-2\)
\(r_{13}\)\(e_{15}\)\(-2\)
\(r_{14}\)\(e_{16}\)\(-2\)
\(r_{15}\)\(e_{17}\)\(-2\)
\(r_{16}\)\(e_{18}\)\(-2\)
\(r_{17}\)\(e_{19}\)\(-2\)
\(r_{18}\)\(e_{20}\)\(-2\)
\(r_{19}\)\(e_{21}\)\(-2\)
\(r_{20}\)\(e_{22}\)\(-2\)
\(r_{21}\)\(e_{23}\)\(-2\)
\(r_{22}\)\(e_{24}\)\(-2\)
\(r_{23}\)\(e_{25}\)\(-2\)
\(r_{24}\)\(e_{26}\)\(-2\)
\(r_{25}\)\(e_{1} + e_{2}\)\(2\)
\(r_{26}\)\(e_{1} + e_{3}\)\(-2\)

Notes

The even unimodular lattice of signature \((1, 25)\), \(U \oplus E_8(-1)^3\), the lattice \(\Gamma\) of Huybrechts (Section 2.2).

Morphisms

References

  1. D. Huybrechts, "On derived categories of K3 surfaces, symplectic automorphisms and the Conway group", arXiv:1309.6528, Section 1.1.