Tag 029K

\(U^{2} \oplus E_{8}(-1)^{3}\)

U^2 + E8(-1)^3·II_{2,26}

Invariants

Rank	28
Signature \((n_+, n_-)\)	(2, 26)
Determinant	\(1\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{28}\).

In this basis, \(L = {}\)\(\mathbb{Z}\{e_{1}, e_{2}\}\) \(\oplus\) \(\mathbb{Z}\{e_{3}, e_{4}\}\) \(\oplus\) \(\mathbb{Z}\{e_{5}, \dots, e_{12}\}\) \(\oplus\) \(\mathbb{Z}\{e_{13}, \dots, e_{20}\}\) \(\oplus\) \(\mathbb{Z}\{e_{21}, \dots, e_{28}\}\), an orthogonal sum.

	\(e_{1}\)	\(e_{2}\)	\(e_{3}\)	\(e_{4}\)	\(e_{5}\)	\(e_{6}\)	\(e_{7}\)	\(e_{8}\)	\(e_{9}\)	\(e_{10}\)	\(e_{11}\)	\(e_{12}\)	\(e_{13}\)	\(e_{14}\)	\(e_{15}\)	\(e_{16}\)	\(e_{17}\)	\(e_{18}\)	\(e_{19}\)	\(e_{20}\)	\(e_{21}\)	\(e_{22}\)	\(e_{23}\)	\(e_{24}\)	\(e_{25}\)	\(e_{26}\)	\(e_{27}\)	\(e_{28}\)
\(e_{1}\)	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
\(e_{2}\)	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	0	0	0
\(e_{3}\)	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
\(e_{4}\)	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
\(e_{5}\)	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
\(e_{6}\)	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
\(e_{7}\)	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
\(e_{8}\)	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
\(e_{9}\)	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
\(e_{10}\)	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
\(e_{11}\)	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
\(e_{12}\)	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
\(e_{13}\)	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
\(e_{14}\)	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
\(e_{15}\)	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
\(e_{16}\)	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
\(e_{17}\)	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
\(e_{18}\)	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
\(e_{19}\)	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
\(e_{20}\)	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
\(e_{21}\)	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
\(e_{22}\)	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
\(e_{23}\)	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
\(e_{24}\)	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
\(e_{25}\)	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
\(e_{26}\)	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
\(e_{27}\)	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	1	-2	1
\(e_{28}\)	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	1	-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, 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], [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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 0, 0, 1, -2]]

Integral lattice

Parity	even
Discriminant group	\(0\)
Unimodular	yes
Integral overlattices \(L \subseteq M \subseteq L^*\)	1
Nikulin invariants \((r, a, \delta)\)	\((28, 0, 0)\)
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)	\(\{2\}\)
Character of the discriminant \(\chi_D\), \(D = (-1)^{14} \det L\)	trivial
Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)	\(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 27)\, \zeta^\Sigma(s - 14)}{\zeta^\Sigma(s - 13)}\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)	\(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 27)}{\zeta^\Sigma(s - 13)}\)
Hyperbolic index \(\max\{n : L \cong U^n \oplus L'\}\)	\(\geq 2\), by an embedding of \(U^{2}\) in the morphism files

Orbits of primitive vectors \(F_{L,\Gamma}(z, w) = \sum_n c_\Gamma(n)\, z^n\), with \(w^n\) for \(z^{-n}\)

\(\Gamma = \mathrm{O}(L)\)	\(1 + z^{2} + w^{2} + z^{4} + w^{4} + O(z^{5}, w^{5})\), by V. Gritsenko, K. Hulek and G. K. Sankaran, "Moduli of K3 surfaces and irreducible symplectic manifolds", arXiv:1012.4155, Lemma 7.5 (Eichler criterion): L contains U^2 and A_L = 0, so the SOtilde-orbit of a primitive vector is determined by its norm; each group contains SOtilde.
\(\Gamma = \mathrm{SO}(L)\)	\(1 + z^{2} + w^{2} + z^{4} + w^{4} + O(z^{5}, w^{5})\), by V. Gritsenko, K. Hulek and G. K. Sankaran, "Moduli of K3 surfaces and irreducible symplectic manifolds", arXiv:1012.4155, Lemma 7.5 (Eichler criterion): L contains U^2 and A_L = 0, so the SOtilde-orbit of a primitive vector is determined by its norm; each group contains SOtilde.
\(\Gamma = \widetilde{O}(L)\)	\(1 + z^{2} + w^{2} + z^{4} + w^{4} + O(z^{5}, w^{5})\), by V. Gritsenko, K. Hulek and G. K. Sankaran, "Moduli of K3 surfaces and irreducible symplectic manifolds", arXiv:1012.4155, Lemma 7.5 (Eichler criterion): L contains U^2 and A_L = 0, so the SOtilde-orbit of a primitive vector is determined by its norm; each group contains SOtilde.
\(\Gamma = \widetilde{SO}(L)\)	\(1 + z^{2} + w^{2} + z^{4} + w^{4} + O(z^{5}, w^{5})\), by V. Gritsenko, K. Hulek and G. K. Sankaran, "Moduli of K3 surfaces and irreducible symplectic manifolds", arXiv:1012.4155, Lemma 7.5 (Eichler criterion): L contains U^2 and A_L = 0, so the SOtilde-orbit of a primitive vector is determined by its norm; each group contains SOtilde.

Indefinite lattice

Isotropic	yes

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 \(R(L)\)	28
\([L : R(L)]\)	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	\(b(r, r)\)
\(r_{1}\)	\(e_{5}\)	\(-2\)
\(r_{2}\)	\(e_{6}\)	\(-2\)
\(r_{3}\)	\(e_{7}\)	\(-2\)
\(r_{4}\)	\(e_{8}\)	\(-2\)
\(r_{5}\)	\(e_{9}\)	\(-2\)
\(r_{6}\)	\(e_{10}\)	\(-2\)
\(r_{7}\)	\(e_{11}\)	\(-2\)
\(r_{8}\)	\(e_{12}\)	\(-2\)
\(r_{9}\)	\(e_{13}\)	\(-2\)
\(r_{10}\)	\(e_{14}\)	\(-2\)
\(r_{11}\)	\(e_{15}\)	\(-2\)
\(r_{12}\)	\(e_{16}\)	\(-2\)
\(r_{13}\)	\(e_{17}\)	\(-2\)
\(r_{14}\)	\(e_{18}\)	\(-2\)
\(r_{15}\)	\(e_{19}\)	\(-2\)
\(r_{16}\)	\(e_{20}\)	\(-2\)
\(r_{17}\)	\(e_{21}\)	\(-2\)
\(r_{18}\)	\(e_{22}\)	\(-2\)
\(r_{19}\)	\(e_{23}\)	\(-2\)
\(r_{20}\)	\(e_{24}\)	\(-2\)
\(r_{21}\)	\(e_{25}\)	\(-2\)
\(r_{22}\)	\(e_{26}\)	\(-2\)
\(r_{23}\)	\(e_{27}\)	\(-2\)
\(r_{24}\)	\(e_{28}\)	\(-2\)
\(r_{25}\)	\(e_{1} + e_{2}\)	\(2\)
\(r_{26}\)	\(e_{1} + e_{5}\)	\(-2\)
\(r_{27}\)	\(e_{3} + e_{4}\)	\(2\)
\(r_{28}\)	\(e_{3} + e_{5}\)	\(-2\)

Notes

The even unimodular lattice \(L_{2,26} = 2U \oplus 3E_8(-1)\) of signature \((2, 26)\). The Borcherds form \(\Phi_{12}\) is the unique modular form of weight 12 and character \(\det\) for \(O^+(L_{2,26})\), and its quasi-pullbacks along embeddings \(L_{2d} \hookrightarrow L_{2,26}\) give cusp forms on the moduli of polarized K3 surfaces (Gritsenko, Hulek and Sankaran, Section 6).

Morphisms

References

V. Gritsenko, K. Hulek, G. K. Sankaran, "The Kodaira dimension of the moduli of K3 surfaces", arXiv:math/0607339, Section 6.