Collection
Lattices that are not integral
Lattices whose form has a value outside the integers.
The form of each member has a component \(b(e_i, e_j)\) that is not an integer, so the lattice is not integral. Parity, the discriminant group, the genus symbol, the theta series and the type of \(\Phi_{\{2\}}(L)\) are undefined for these lattices.
39 lattices
This collection in the database
| Tag | Lattice | Rank | Signature | Determinant | Properties |
|---|---|---|---|---|---|
| 0006 | \(\langle \frac{1}{2} \rangle\) | 1 | (1, 0) | 1/2 | not integral root lattice |
| 0017 | \(A_{2}^{*}\) | 2 | (2, 0) | 1/3 | not integral root lattice |
| 0027 | \(A_{3}^{*}\) | 3 | (3, 0) | 1/4 | not integral not a root lattice |
| 0041 | \(A_{4}^{*}\) | 4 | (4, 0) | 1/5 | not integral not a root lattice |
| 0042 | \(D_{4}^{*}\) | 4 | (4, 0) | 1/4 | not integral root lattice |
| 0054 | \(A_{5}^{*}\) | 5 | (5, 0) | 1/6 | not integral not a root lattice |
| 0055 | \(D_{5}^{*}\) | 5 | (5, 0) | 1/4 | not integral not a root lattice |
| 0068 | \(A_{6}^{*}\) | 6 | (6, 0) | 1/7 | not integral not a root lattice |
| 0069 | \(D_{6}^{*}\) | 6 | (6, 0) | 1/4 | not integral not a root lattice |
| 0070 | \(E_{6}^{*}\) | 6 | (6, 0) | 1/3 | not integral not a root lattice |
| 0083 | \(A_{7}^{*}\) | 7 | (7, 0) | 1/8 | not integral not a root lattice |
| 0084 | \(D_{7}^{*}\) | 7 | (7, 0) | 1/4 | not integral not a root lattice |
| 0085 | \(E_{7}^{*}\) | 7 | (7, 0) | 1/2 | not integral not a root lattice |
| 0100 | \(A_{8}^{*}\) | 8 | (8, 0) | 1/9 | not integral not a root lattice |
| 0101 | \(D_{8}^{*}\) | 8 | (8, 0) | 1/4 | not integral not a root lattice |
| 0114 | \(A_{9}^{*}\) | 9 | (9, 0) | 1/10 | not integral not a root lattice |
| 0115 | \(D_{9}^{*}\) | 9 | (9, 0) | 1/4 | not integral not a root lattice |
| 0129 | \(A_{10}^{*}\) | 10 | (10, 0) | 1/11 | not integral not a root lattice |
| 0130 | \(D_{10}^{*}\) | 10 | (10, 0) | 1/4 | not integral not a root lattice |
| 0141 | \(A_{11}^{*}\) | 11 | (11, 0) | 1/12 | not integral not a root lattice |
| 0142 | \(D_{11}^{*}\) | 11 | (11, 0) | 1/4 | not integral not a root lattice |
| 0157 | \(A_{12}^{*}\) | 12 | (12, 0) | 1/13 | not integral not a root lattice |
| 0158 | \(D_{12}^{*}\) | 12 | (12, 0) | 1/4 | not integral not a root lattice |
| 0170 | \(A_{13}^{*}\) | 13 | (13, 0) | 1/14 | not integral not a root lattice |
| 0171 | \(D_{13}^{*}\) | 13 | (13, 0) | 1/4 | not integral not a root lattice |
| 0184 | \(A_{14}^{*}\) | 14 | (14, 0) | 1/15 | not integral not a root lattice |
| 0185 | \(D_{14}^{*}\) | 14 | (14, 0) | 1/4 | not integral not a root lattice |
| 0197 | \(A_{15}^{*}\) | 15 | (15, 0) | 1/16 | not integral not a root lattice |
| 0198 | \(D_{15}^{*}\) | 15 | (15, 0) | 1/4 | not integral not a root lattice |
| 0211 | \(A_{16}^{*}\) | 16 | (16, 0) | 1/17 | not integral not a root lattice |
| 0212 | \(D_{16}^{*}\) | 16 | (16, 0) | 1/4 | not integral not a root lattice |
| 0223 | \(A_{17}^{*}\) | 17 | (17, 0) | 1/18 | not integral not a root lattice |
| 0224 | \(D_{17}^{*}\) | 17 | (17, 0) | 1/4 | not integral not a root lattice |
| 0237 | \(A_{18}^{*}\) | 18 | (18, 0) | 1/19 | not integral not a root lattice |
| 0238 | \(D_{18}^{*}\) | 18 | (18, 0) | 1/4 | not integral not a root lattice |
| 0249 | \(A_{19}^{*}\) | 19 | (19, 0) | 1/20 | not integral not a root lattice |
| 0250 | \(D_{19}^{*}\) | 19 | (19, 0) | 1/4 | not integral not a root lattice |
| 0262 | \(A_{20}^{*}\) | 20 | (20, 0) | 1/21 | not integral not a root lattice |
| 0263 | \(D_{20}^{*}\) | 20 | (20, 0) | 1/4 | not integral not a root lattice |