Tag 0138
\(\langle 2 \rangle \oplus A_{1}(-1)^{10}\)
<2> + A1(-1)^10·(r, a, delta) = (11, 11, 1)
Invariants
| Rank | 11 |
|---|---|
| Signature \((n_+, n_-)\) | (1, 10) |
| Determinant | \(2048\) |
Gram tensor
The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{11}\).
In this basis, \(L = {}\)\(\mathbb{Z}\{e_{1}\}\) \(\oplus\) \(\mathbb{Z}\{e_{2}\}\) \(\oplus\) \(\mathbb{Z}\{e_{3}\}\) \(\oplus\) \(\mathbb{Z}\{e_{4}\}\) \(\oplus\) \(\mathbb{Z}\{e_{5}\}\) \(\oplus\) \(\mathbb{Z}\{e_{6}\}\) \(\oplus\) \(\mathbb{Z}\{e_{7}\}\) \(\oplus\) \(\mathbb{Z}\{e_{8}\}\) \(\oplus\) \(\mathbb{Z}\{e_{9}\}\) \(\oplus\) \(\mathbb{Z}\{e_{10}\}\) \(\oplus\) \(\mathbb{Z}\{e_{11}\}\), an orthogonal sum.
| \(e_{1}\) | \(e_{2}\) | \(e_{3}\) | \(e_{4}\) | \(e_{5}\) | \(e_{6}\) | \(e_{7}\) | \(e_{8}\) | \(e_{9}\) | \(e_{10}\) | \(e_{11}\) | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| \(e_{1}\) | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(e_{2}\) | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(e_{3}\) | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(e_{4}\) | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(e_{5}\) | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(e_{6}\) | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 |
| \(e_{7}\) | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 |
| \(e_{8}\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 |
| \(e_{9}\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 |
| \(e_{10}\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 |
| \(e_{11}\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 |
[[2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2]]
Integral lattice
| Parity | even |
|---|---|
| Discriminant group | \((\mathbb{Z}/2)^{11}\) |
| Unimodular | no |
| Integral overlattices \(L \subseteq M \subseteq L^*\) | not decided |
| Nikulin invariants \((r, a, \delta)\) | \((11, 11, 1)\) |
| Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\) | \(\{2\}\) |
| Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\) | \(\zeta^\Sigma(X_0, s) = \zeta^\Sigma(s - 10)\) |
| Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\) | \(\zeta^\Sigma(X_n, s) = \zeta^\Sigma(s - 10)\, L^\Sigma(s - 5, \chi_{-2048n})\) |
| Genus symbol | \(\mathrm{II}_{1,10}\ \bigl(2\colon [2^{11}]_{7}\bigr)\) |
| Class number of the genus | 1 |
Indefinite lattice
| Isotropic | yes |
|---|
Hyperbolic lattice
| Reflective | yes 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 \(R(L)\) | 11 |
| \([L : R(L)]\) | 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 | \(b(r, r)\) |
|---|---|---|
| \(r_{1}\) | \(e_{1}\) | \(2\) |
| \(r_{2}\) | \(e_{2}\) | \(-2\) |
| \(r_{3}\) | \(e_{3}\) | \(-2\) |
| \(r_{4}\) | \(e_{4}\) | \(-2\) |
| \(r_{5}\) | \(e_{5}\) | \(-2\) |
| \(r_{6}\) | \(e_{6}\) | \(-2\) |
| \(r_{7}\) | \(e_{7}\) | \(-2\) |
| \(r_{8}\) | \(e_{8}\) | \(-2\) |
| \(r_{9}\) | \(e_{9}\) | \(-2\) |
| \(r_{10}\) | \(e_{10}\) | \(-2\) |
| \(r_{11}\) | \(e_{11}\) | \(-2\) |
Notes
This is the even hyperbolic 2-elementary lattice with invariants \((r, a, \delta) = (11, 11, 1)\).
Since \(a = r\), \(L^* = \tfrac{1}{2} L\), and \(L\) is the twist \(M(2)\) of the unimodular lattice \(M = \mathrm{I}_{1,10}\), the record 0139. The corpus records a twist \(M(n)\), \(|n| \geq 2\), only for the 13 rows \(a = r\) of Nikulin’s table, because the classification names the twist and not \(M\).
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 reflective (Alexeev, Theorem 1.1).
\(\langle a \rangle\) is the lattice of rank 1 with basis \(e\) and \(b(e, e) = a\), and \(A_1(-1) = \langle -2 \rangle\). The basis of an orthogonal sum is the union of the bases of the summands, in the order written.
Related lattices
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