Collection
Even hyperbolic 2-elementary lattices with \(r + a = 22\)
The rows of Nikulin's Table 1 with \(r + a = 22\), with the reflectivity of each from Alexeev's Theorem 1.1.
The members of this collection are the even hyperbolic 2-elementary lattices with \(r + a = 22\), for Nikulin’s invariants \((r, a, \delta)\). They are rows of Table 1 of 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, DOI 10.1007/BF01094757, and they are members of the collection Nikulin’s 2-elementary lattices.
The reflectivity of each record is from V. Alexeev, Reflective hyperbolic 2-elementary lattices, K3 surfaces and hyperkahler manifolds, arXiv:2209.09110v4, Theorem 1.1: the lattices with \((r, a, \delta)\) equal to \((17, 5, 1)\), \((18, 4, 1)\) and \((19, 3, 1)\) are not reflective, and the other lattices with \(r + a = 22\) are reflective.
12 lattices
This collection in the database
| Tag | Lattice | Rank | Signature | Determinant | Properties |
|---|---|---|---|---|---|
| 0138 | \(\langle 2 \rangle \oplus A_{1}(-1)^{10}\) | 11 | (1, 10) | 2048 | integral even 2-elementary root lattice hyperbolic isotropic reflective |
| 0152 | \(U \oplus A_{1}(-1)^{10}\) | 12 | (1, 11) | -1024 | integral even 2-elementary root lattice hyperbolic isotropic reflective |
| 0167 | \(U \oplus D_{4}(-1) \oplus A_{1}(-1)^{7}\) | 13 | (1, 12) | 512 | integral even 2-elementary root lattice hyperbolic isotropic reflective |
| 0179 | \(U(2) \oplus D_{4}(-1)^{3}\) | 14 | (1, 13) | -256 | integral even 2-elementary root lattice hyperbolic isotropic reflective |
| 0180 | \(U \oplus D_{4}(-1)^{2} \oplus A_{1}(-1)^{4}\) | 14 | (1, 13) | -256 | integral even 2-elementary root lattice hyperbolic isotropic reflective |
| 0194 | \(U \oplus E_{7}(-1) \oplus A_{1}(-1)^{6}\) | 15 | (1, 14) | 128 | integral even 2-elementary root lattice hyperbolic isotropic reflective |
| 0207 | \(U \oplus E_{8}(-1) \oplus A_{1}(-1)^{6}\) | 16 | (1, 15) | -64 | integral even 2-elementary root lattice hyperbolic isotropic reflective |
| 0220 | \(U \oplus E_{8}(-1) \oplus D_{4}(-1) \oplus A_{1}(-1)^{3}\) | 17 | (1, 16) | 32 | integral even 2-elementary root lattice hyperbolic isotropic not reflective |
| 0232 | \(U \oplus E_{8}(-1) \oplus D_{4}(-1)^{2}\) | 18 | (1, 17) | -16 | integral even 2-elementary root lattice hyperbolic isotropic reflective |
| 0233 | \(U \oplus E_{8}(-1) \oplus D_{6}(-1) \oplus A_{1}(-1)^{2}\) | 18 | (1, 17) | -16 | integral even 2-elementary root lattice hyperbolic isotropic not reflective |
| 0246 | \(U \oplus E_{8}(-1) \oplus E_{7}(-1) \oplus A_{1}(-1)^{2}\) | 19 | (1, 18) | 8 | integral even 2-elementary root lattice hyperbolic isotropic not reflective |
| 0259 | \(U \oplus E_{8}(-1)^{2} \oplus A_{1}(-1)^{2}\) | 20 | (1, 19) | -4 | integral even 2-elementary root lattice hyperbolic isotropic reflective |