Tag 0236

\(U \oplus E_{8}(-1)^{2}\)

U + E8(-1)^2·II_{1,17}·(r, a, delta) = (18, 0, 0)

Invariants

Rank18
Signature \((n_+, n_-)\)(1, 17)
Determinant\(-1\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{18}\).

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

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)\(e_{13}\)\(e_{14}\)\(e_{15}\)\(e_{16}\)\(e_{17}\)\(e_{18}\)
\(e_{1}\)010000000000000000
\(e_{2}\)100000000000000000
\(e_{3}\)00-2010000000000000
\(e_{4}\)000-201000000000000
\(e_{5}\)0010-21000000000000
\(e_{6}\)00011-2100000000000
\(e_{7}\)000001-210000000000
\(e_{8}\)0000001-21000000000
\(e_{9}\)00000001-2100000000
\(e_{10}\)000000001-200000000
\(e_{11}\)0000000000-20100000
\(e_{12}\)00000000000-2010000
\(e_{13}\)000000000010-210000
\(e_{14}\)0000000000011-21000
\(e_{15}\)00000000000001-2100
\(e_{16}\)000000000000001-210
\(e_{17}\)0000000000000001-21
\(e_{18}\)00000000000000001-2
 
[[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, 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, -2, 0, 1, 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, 1, 1, -2, 1, 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, 1, -2, 1, 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, 1, -2, 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, -2, 0, 1, 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, 1, 1, -2, 1, 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, 1, -2, 1, 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, 1, -2]]

Integral lattice

Parityeven
Discriminant group\(0\)
Unimodularyes
Integral overlattices \(L \subseteq M \subseteq L^*\)1
Nikulin invariants \((r, a, \delta)\)\((18, 0, 0)\)
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)\(\{2\}\)
Character of the discriminant \(\chi_D\), \(D = (-1)^{9} \det L\)trivial
Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)\(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 17)\, \zeta^\Sigma(s - 9)}{\zeta^\Sigma(s - 8)}\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)\(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 17)}{\zeta^\Sigma(s - 8)}\)
Genus symbol\(\mathrm{II}_{1,17}\)
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 \(R(L)\)18
\([L : R(L)]\)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\(b(r, r)\)
\(r_{1}\)\(e_{3}\)\(-2\)
\(r_{2}\)\(e_{4}\)\(-2\)
\(r_{3}\)\(e_{5}\)\(-2\)
\(r_{4}\)\(e_{6}\)\(-2\)
\(r_{5}\)\(e_{7}\)\(-2\)
\(r_{6}\)\(e_{8}\)\(-2\)
\(r_{7}\)\(e_{9}\)\(-2\)
\(r_{8}\)\(e_{10}\)\(-2\)
\(r_{9}\)\(e_{11}\)\(-2\)
\(r_{10}\)\(e_{12}\)\(-2\)
\(r_{11}\)\(e_{13}\)\(-2\)
\(r_{12}\)\(e_{14}\)\(-2\)
\(r_{13}\)\(e_{15}\)\(-2\)
\(r_{14}\)\(e_{16}\)\(-2\)
\(r_{15}\)\(e_{17}\)\(-2\)
\(r_{16}\)\(e_{18}\)\(-2\)
\(r_{17}\)\(e_{1} + e_{2}\)\(2\)
\(r_{18}\)\(e_{1} + e_{3}\)\(-2\)

Notes

This is an even hyperbolic unimodular lattice, so \(A_L = 0\) and its invariants are \((r, a, \delta) = (18, 0, 0)\).

Nikulin lists it in Table 1 of Factor groups of groups of automorphisms of hyperbolic forms with respect to subgroups generated by 2-reflections (1983), the table of the even hyperbolic lattices \(L\) with \(2 A_L = 0\) that have a primitive embedding in the K3 lattice. These are the invariant lattices of the nonsymplectic involutions of K3 surfaces (Remark 4.5.3), and Theorem 4.3.2 states that \((r, a, \delta)\) determines each of them.

Morphisms

References

  1. V. V. Nikulin, "Factor groups of groups of automorphisms of hyperbolic forms with respect to subgroups generated by 2-reflections. Algebrogeometric applications", J. Soviet Math. 22 (1983), 1401-1475, §4, Table 1 and Theorem 4.3.2. https://doi.org/10.1007/BF01094757