Tag 0220

\(U \oplus E_{8}(-1) \oplus D_{4}(-1) \oplus A_{1}(-1)^{3}\)

U + E8(-1) + D4(-1) + A1(-1)^3·(r, a, delta) = (17, 5, 1)

Invariants

Rank	17
Signature \((n_+, n_-)\)	(1, 16)
Determinant	\(32\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{17}\).

In this basis, \(L = {}\)\(\mathbb{Z}\{e_{1}, e_{2}\}\) \(\oplus\) \(\mathbb{Z}\{e_{3}, \dots, e_{10}\}\) \(\oplus\) \(\mathbb{Z}\{e_{11}, \dots, e_{14}\}\) \(\oplus\) \(\mathbb{Z}\{e_{15}\}\) \(\oplus\) \(\mathbb{Z}\{e_{16}\}\) \(\oplus\) \(\mathbb{Z}\{e_{17}\}\), an orthogonal sum.

	\(e_{1}\)	\(e_{2}\)	\(e_{3}\)	\(e_{4}\)	\(e_{5}\)	\(e_{6}\)	\(e_{7}\)	\(e_{8}\)	\(e_{9}\)	\(e_{10}\)	\(e_{11}\)	\(e_{12}\)	\(e_{13}\)	\(e_{14}\)	\(e_{15}\)	\(e_{16}\)	\(e_{17}\)
\(e_{1}\)	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
\(e_{2}\)	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
\(e_{3}\)	0	0	-2	0	1	0	0	0	0	0	0	0	0	0	0	0	0
\(e_{4}\)	0	0	0	-2	0	1	0	0	0	0	0	0	0	0	0	0	0
\(e_{5}\)	0	0	1	0	-2	1	0	0	0	0	0	0	0	0	0	0	0
\(e_{6}\)	0	0	0	1	1	-2	1	0	0	0	0	0	0	0	0	0	0
\(e_{7}\)	0	0	0	0	0	1	-2	1	0	0	0	0	0	0	0	0	0
\(e_{8}\)	0	0	0	0	0	0	1	-2	1	0	0	0	0	0	0	0	0
\(e_{9}\)	0	0	0	0	0	0	0	1	-2	1	0	0	0	0	0	0	0
\(e_{10}\)	0	0	0	0	0	0	0	0	1	-2	0	0	0	0	0	0	0
\(e_{11}\)	0	0	0	0	0	0	0	0	0	0	-2	1	0	0	0	0	0
\(e_{12}\)	0	0	0	0	0	0	0	0	0	0	1	-2	1	1	0	0	0
\(e_{13}\)	0	0	0	0	0	0	0	0	0	0	0	1	-2	0	0	0	0
\(e_{14}\)	0	0	0	0	0	0	0	0	0	0	0	1	0	-2	0	0	0
\(e_{15}\)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	-2	0	0
\(e_{16}\)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	-2	0
\(e_{17}\)	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], [1, 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, -2, 0, 1, 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, 1, 1, -2, 1, 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, 1, -2, 1, 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, 1, -2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1, 1, 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, 1, 0, -2, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2]]

Integral lattice

Parity	even
Discriminant group	\((\mathbb{Z}/2)^{5}\)
Unimodular	no
Integral overlattices \(L \subseteq M \subseteq L^*\)	31
Nikulin invariants \((r, a, \delta)\)	\((17, 5, 1)\)
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)	\(\{2\}\)
Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)	\(\zeta^\Sigma(X_0, s) = \zeta^\Sigma(s - 16)\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)	\(\zeta^\Sigma(X_n, s) = \zeta^\Sigma(s - 16)\, L^\Sigma(s - 8, \chi_{32n})\)
Genus symbol	\(\mathrm{II}_{1,16}\ \bigl(2\colon 1^{12}\,[2^{5}]_{1}\bigr)\)
Hyperbolic index \(\max\{n : L \cong U^n \oplus L'\}\)	\(\geq 1\), by an embedding of \(U^{1}\) in the morphism files

Indefinite lattice

Isotropic	yes

Hyperbolic lattice

Reflective	no declared

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 \(R(L)\)	17
\([L : R(L)]\)	1
\(R(L)\) is primitive in \(L\)	yes
Proof	Each \(r_i\) 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 \(L\).

Root	Coordinates	\(b(r, r)\)
\(r_{1}\)	\(e_{3}\)	\(-2\)
\(r_{2}\)	\(e_{4}\)	\(-2\)
\(r_{3}\)	\(e_{5}\)	\(-2\)
\(r_{4}\)	\(e_{6}\)	\(-2\)
\(r_{5}\)	\(e_{7}\)	\(-2\)
\(r_{6}\)	\(e_{8}\)	\(-2\)
\(r_{7}\)	\(e_{9}\)	\(-2\)
\(r_{8}\)	\(e_{10}\)	\(-2\)
\(r_{9}\)	\(e_{11}\)	\(-2\)
\(r_{10}\)	\(e_{12}\)	\(-2\)
\(r_{11}\)	\(e_{13}\)	\(-2\)
\(r_{12}\)	\(e_{14}\)	\(-2\)
\(r_{13}\)	\(e_{15}\)	\(-2\)
\(r_{14}\)	\(e_{16}\)	\(-2\)
\(r_{15}\)	\(e_{17}\)	\(-2\)
\(r_{16}\)	\(e_{1} + e_{2}\)	\(2\)
\(r_{17}\)	\(e_{1} + e_{3}\)	\(-2\)

Notes

This is the even hyperbolic 2-elementary lattice with invariants \((r, a, \delta) = (17, 5, 1)\).

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.

Its invariants satisfy \(r + a = 22\), and it is not reflective (Alexeev, Theorem 1.1).

Morphisms

References

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
V. Alexeev, "Reflective hyperbolic 2-elementary lattices, K3 surfaces and hyperkahler manifolds", arXiv:2209.09110v4, Theorem 1.1. https://arxiv.org/abs/2209.09110v4