Collection
2-elementary lattices
Integral lattices with discriminant group \((\mathbb{Z}/2)^a\), \(a \geq 1\).
A nondegenerate integral lattice is 2-elementary when its discriminant group is isomorphic to \((\mathbb{Z}/2)^a\) with \(a \geq 1\).
106 lattices
This collection in the database
| Tag | Lattice | Rank | Signature | Determinant | Properties |
|---|---|---|---|---|---|
| 0002 | \(\langle 2 \rangle\) | 1 | (1, 0) | 2 | integral even 2-elementary root lattice |
| 0013 | \(U(2)\) | 2 | (1, 1) | -4 | integral even 2-elementary not a root lattice hyperbolic isotropic |
| 0014 | \(\langle 2 \rangle \oplus \langle -2 \rangle\) | 2 | (1, 1) | -4 | integral even 2-elementary root lattice hyperbolic isotropic |
| 0021 | \(\langle 1 \rangle \oplus \langle -2 \rangle\) | 2 | (1, 1) | -2 | integral odd 2-elementary root lattice hyperbolic anisotropic |
| 0024 | \(U(2) \oplus \langle -2 \rangle\) | 3 | (1, 2) | 8 | integral even 2-elementary root lattice hyperbolic isotropic |
| 0026 | \(U \oplus A_{1}(-1)\) | 3 | (1, 2) | 2 | integral even 2-elementary root lattice hyperbolic isotropic |
| 0035 | \(D_{4}\) | 4 | (4, 0) | 4 | integral even 2-elementary root lattice |
| 0036 | \(U \oplus U(2)\) | 4 | (2, 2) | 4 | integral even 2-elementary not a root lattice isotropic |
| 0057 | \(D_{4} \oplus A_{1}\) | 5 | (5, 0) | 8 | integral even 2-elementary root lattice |
| 0063 | \(D_{6}\) | 6 | (6, 0) | 4 | integral even 2-elementary root lattice |
| 0066 | \(U \oplus D_{4}(-1)\) | 6 | (1, 5) | -4 | integral even 2-elementary root lattice hyperbolic isotropic |
| 0079 | \(E_{7}\) | 7 | (7, 0) | 2 | integral even 2-elementary root lattice |
| 0093 | \(D_{8}\) | 8 | (8, 0) | 4 | integral even 2-elementary root lattice |
| 0102 | \(D_{4}^{2}\) | 8 | (8, 0) | 16 | integral even 2-elementary root lattice |
| 0103 | \(E_{7} \oplus A_{1}\) | 8 | (8, 0) | 4 | integral even 2-elementary root lattice |
| 0113 | \(U \oplus E_{7}(-1)\) | 9 | (1, 8) | 2 | integral even 2-elementary root lattice hyperbolic isotropic |
| 0116 | \(E_{8} \oplus A_{1}\) | 9 | (9, 0) | 2 | integral even 2-elementary root lattice |
| 0122 | \(D_{10}\) | 10 | (10, 0) | 4 | integral even 2-elementary root lattice |
| 0124 | \(U \oplus E_{8}(-2)\) | 10 | (1, 9) | -256 | integral even 2-elementary not a root lattice hyperbolic isotropic |
| 0125 | \(U(2) \oplus E_{8}(-2)\) | 10 | (1, 9) | -1024 | integral even 2-elementary root lattice hyperbolic isotropic |
| 0127 | \(U \oplus D_{8}(-1)\) | 10 | (1, 9) | -4 | integral even 2-elementary root lattice hyperbolic isotropic |
| 0138 | \(\langle 2 \rangle \oplus A_{1}(-1)^{10}\) | 11 | (1, 10) | 2048 | integral even 2-elementary root lattice hyperbolic isotropic reflective |
| 0148 | \(D_{12}\) | 12 | (12, 0) | 4 | integral even 2-elementary root lattice |
| 0152 | \(U \oplus A_{1}(-1)^{10}\) | 12 | (1, 11) | -1024 | integral even 2-elementary root lattice hyperbolic isotropic reflective |
| 0153 | \(U^{2} \oplus E_{8}(-2)\) | 12 | (2, 10) | 256 | integral even 2-elementary root lattice isotropic |
| 0155 | \(U \oplus D_{10}(-1)\) | 12 | (1, 11) | -4 | integral even 2-elementary root lattice hyperbolic isotropic |
| 0159 | \(E_{8} \oplus D_{4}\) | 12 | (12, 0) | 4 | integral even 2-elementary root lattice |
| 0167 | \(U \oplus D_{4}(-1) \oplus A_{1}(-1)^{7}\) | 13 | (1, 12) | 512 | integral even 2-elementary root lattice hyperbolic isotropic reflective |
| 0177 | \(D_{14}\) | 14 | (14, 0) | 4 | integral even 2-elementary root lattice |
| 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 |
| 0182 | \(U \oplus D_{12}(-1)\) | 14 | (1, 13) | -4 | integral even 2-elementary root lattice hyperbolic isotropic |
| 0187 | \(E_{7}^{2}\) | 14 | (14, 0) | 4 | integral even 2-elementary root lattice |
| 0194 | \(U \oplus E_{7}(-1) \oplus A_{1}(-1)^{6}\) | 15 | (1, 14) | 128 | integral even 2-elementary root lattice hyperbolic isotropic reflective |
| 0199 | \(E_{8} \oplus E_{7}\) | 15 | (15, 0) | 2 | integral even 2-elementary root lattice |
| 0204 | \(D_{16}\) | 16 | (16, 0) | 4 | integral even 2-elementary root lattice |
| 0205 | \(\Lambda_{16}\) | 16 | (16, 0) | 256 | integral even 2-elementary not a root lattice |
| 0207 | \(U \oplus E_{8}(-1) \oplus A_{1}(-1)^{6}\) | 16 | (1, 15) | -64 | integral even 2-elementary root lattice hyperbolic isotropic reflective |
| 0209 | \(U \oplus D_{14}(-1)\) | 16 | (1, 15) | -4 | integral even 2-elementary root lattice hyperbolic isotropic |
| 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 |
| 0225 | \(E_{8}^{2} \oplus A_{1}\) | 17 | (17, 0) | 2 | integral even 2-elementary root lattice |
| 0230 | \(D_{18}\) | 18 | (18, 0) | 4 | integral even 2-elementary root lattice |
| 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 |
| 0235 | \(U \oplus D_{16}(-1)\) | 18 | (1, 17) | -4 | integral even 2-elementary root lattice hyperbolic isotropic |
| 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 |
| 0256 | \(D_{20}\) | 20 | (20, 0) | 4 | integral even 2-elementary root lattice |
| 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 |
| 0264 | \(E_{8}^{2} \oplus D_{4}\) | 20 | (20, 0) | 4 | integral even 2-elementary root lattice |
| 026A | \(U \oplus A_{1}(-1)^{2}\) | 4 | (1, 3) | -4 | integral even 2-elementary root lattice hyperbolic isotropic |
| 026B | \(U \oplus A_{1}(-1)^{3}\) | 5 | (1, 4) | 8 | integral even 2-elementary root lattice hyperbolic isotropic |
| 026C | \(U(2) \oplus D_{4}(-1)\) | 6 | (1, 5) | -16 | integral even 2-elementary root lattice hyperbolic isotropic |
| 026D | \(U \oplus A_{1}(-1)^{4}\) | 6 | (1, 5) | -16 | integral even 2-elementary root lattice hyperbolic isotropic |
| 026E | \(U \oplus D_{4}(-1) \oplus A_{1}(-1)\) | 7 | (1, 6) | 8 | integral even 2-elementary root lattice hyperbolic isotropic |
| 026F | \(U(2) \oplus D_{4}(-1) \oplus A_{1}(-1)\) | 7 | (1, 6) | 32 | integral even 2-elementary root lattice hyperbolic isotropic |
| 026G | \(U \oplus D_{6}(-1)\) | 8 | (1, 7) | -4 | integral even 2-elementary root lattice hyperbolic isotropic |
| 026H | \(U(2) \oplus D_{6}(-1)\) | 8 | (1, 7) | -16 | integral even 2-elementary root lattice hyperbolic isotropic |
| 026I | \(U(2) \oplus D_{4}(-1) \oplus A_{1}(-1)^{2}\) | 8 | (1, 7) | -64 | integral even 2-elementary root lattice hyperbolic isotropic |
| 026J | \(U(2) \oplus E_{7}(-1)\) | 9 | (1, 8) | 8 | integral even 2-elementary root lattice hyperbolic isotropic |
| 026K | \(U(2) \oplus D_{6}(-1) \oplus A_{1}(-1)\) | 9 | (1, 8) | 32 | integral even 2-elementary root lattice hyperbolic isotropic |
| 026L | \(U \oplus A_{1}(-1)^{7}\) | 9 | (1, 8) | 128 | integral even 2-elementary root lattice hyperbolic isotropic |
| 026M | \(U \oplus E_{7}(-1) \oplus A_{1}(-1)\) | 10 | (1, 9) | -4 | integral even 2-elementary root lattice hyperbolic isotropic |
| 026N | \(U(2) \oplus D_{8}(-1)\) | 10 | (1, 9) | -16 | integral even 2-elementary root lattice hyperbolic isotropic |
| 026O | \(U \oplus D_{6}(-1) \oplus A_{1}(-1)^{2}\) | 10 | (1, 9) | -16 | integral even 2-elementary root lattice hyperbolic isotropic |
| 026P | \(U(2) \oplus D_{4}(-1)^{2}\) | 10 | (1, 9) | -64 | integral even 2-elementary root lattice hyperbolic isotropic |
| 026Q | \(U(2) \oplus D_{6}(-1) \oplus A_{1}(-1)^{2}\) | 10 | (1, 9) | -64 | integral even 2-elementary root lattice hyperbolic isotropic |
| 026R | \(U \oplus A_{1}(-1)^{8}\) | 10 | (1, 9) | -256 | integral even 2-elementary root lattice hyperbolic isotropic |
| 026S | \(U \oplus E_{8}(-1) \oplus A_{1}(-1)\) | 11 | (1, 10) | 2 | integral even 2-elementary root lattice hyperbolic isotropic |
| 026T | \(U \oplus D_{8}(-1) \oplus A_{1}(-1)\) | 11 | (1, 10) | 8 | integral even 2-elementary root lattice hyperbolic isotropic |
| 026U | \(U \oplus D_{6}(-1) \oplus A_{1}(-1)^{3}\) | 11 | (1, 10) | 32 | integral even 2-elementary root lattice hyperbolic isotropic |
| 026V | \(U(2) \oplus D_{6}(-1) \oplus A_{1}(-1)^{3}\) | 11 | (1, 10) | 128 | integral even 2-elementary root lattice hyperbolic isotropic |
| 026W | \(U \oplus E_{8}(-2) \oplus A_{1}(-1)\) | 11 | (1, 10) | 512 | integral even 2-elementary root lattice hyperbolic isotropic |
| 026X | \(U \oplus D_{8}(-1) \oplus A_{1}(-1)^{2}\) | 12 | (1, 11) | -16 | integral even 2-elementary root lattice hyperbolic isotropic |
| 026Y | \(U(2) \oplus D_{6}(-1) \oplus D_{4}(-1)\) | 12 | (1, 11) | -64 | integral even 2-elementary root lattice hyperbolic isotropic |
| 026Z | \(U(2) \oplus D_{6}(-1) \oplus A_{1}(-1)^{4}\) | 12 | (1, 11) | -256 | integral even 2-elementary root lattice hyperbolic isotropic |
| 0270 | \(U \oplus E_{7}(-1) \oplus D_{4}(-1)\) | 13 | (1, 12) | 8 | integral even 2-elementary root lattice hyperbolic isotropic |
| 0271 | \(U(2) \oplus E_{7}(-1) \oplus D_{4}(-1)\) | 13 | (1, 12) | 32 | integral even 2-elementary root lattice hyperbolic isotropic |
| 0272 | \(U \oplus D_{6}(-1) \oplus A_{1}(-1)^{5}\) | 13 | (1, 12) | 128 | integral even 2-elementary root lattice hyperbolic isotropic |
| 0273 | \(U \oplus D_{8}(-1) \oplus D_{4}(-1)\) | 14 | (1, 13) | -16 | integral even 2-elementary root lattice hyperbolic isotropic |
| 0274 | \(U \oplus D_{6}(-1)^{2}\) | 14 | (1, 13) | -16 | integral even 2-elementary root lattice hyperbolic isotropic |
| 0275 | \(U(2) \oplus D_{8}(-1) \oplus D_{4}(-1)\) | 14 | (1, 13) | -64 | integral even 2-elementary root lattice hyperbolic isotropic |
| 0276 | \(U(2) \oplus D_{6}(-1)^{2}\) | 14 | (1, 13) | -64 | integral even 2-elementary root lattice hyperbolic isotropic |
| 0277 | \(U \oplus E_{7}(-1) \oplus D_{6}(-1)\) | 15 | (1, 14) | 8 | integral even 2-elementary root lattice hyperbolic isotropic |
| 0278 | \(U(2) \oplus E_{7}(-1) \oplus D_{6}(-1)\) | 15 | (1, 14) | 32 | integral even 2-elementary root lattice hyperbolic isotropic |
| 0279 | \(U \oplus D_{8}(-1) \oplus D_{6}(-1)\) | 16 | (1, 15) | -16 | integral even 2-elementary root lattice hyperbolic isotropic |
| 027A | \(U \oplus E_{8}(-1) \oplus E_{7}(-1)\) | 17 | (1, 16) | 2 | integral even 2-elementary root lattice hyperbolic isotropic |
| 027B | \(U \oplus E_{7}(-1) \oplus D_{8}(-1)\) | 17 | (1, 16) | 8 | integral even 2-elementary root lattice hyperbolic isotropic |
| 027C | \(U \oplus E_{8}(-1) \oplus E_{7}(-1) \oplus A_{1}(-1)\) | 18 | (1, 17) | -4 | integral even 2-elementary root lattice hyperbolic isotropic |
| 027D | \(U \oplus E_{8}(-1)^{2} \oplus A_{1}(-1)\) | 19 | (1, 18) | 2 | integral even 2-elementary root lattice hyperbolic isotropic |
| 027Z | \(U^{3} \oplus E_{8}(-1)^{2} \oplus \langle -2 \rangle\) | 23 | (3, 20) | 2 | integral even 2-elementary root lattice isotropic |
| 0287 | \(U^{3} \oplus \langle -2 \rangle^{2}\) | 8 | (3, 5) | -4 | integral even 2-elementary root lattice isotropic |
| 0289 | \(U^{2} \oplus E_{8}(-1)^{2} \oplus \langle -2 \rangle\) | 21 | (2, 19) | -2 | integral even 2-elementary root lattice isotropic |
| 028L | \(\langle 2 \rangle \oplus A_{1}(-1)^{3}\) | 4 | (1, 3) | -16 | integral even 2-elementary root lattice hyperbolic isotropic |
| 028M | \(\langle 2 \rangle \oplus A_{1}(-1)^{4}\) | 5 | (1, 4) | 32 | integral even 2-elementary root lattice hyperbolic isotropic |
| 028N | \(\langle 2 \rangle \oplus A_{1}(-1)^{5}\) | 6 | (1, 5) | -64 | integral even 2-elementary root lattice hyperbolic isotropic |
| 028O | \(\langle 2 \rangle \oplus A_{1}(-1)^{6}\) | 7 | (1, 6) | 128 | integral even 2-elementary root lattice hyperbolic isotropic |
| 028P | \(\langle 2 \rangle \oplus A_{1}(-1)^{7}\) | 8 | (1, 7) | -256 | integral even 2-elementary root lattice hyperbolic isotropic |
| 028Q | \(\langle 2 \rangle \oplus A_{1}(-1)^{8}\) | 9 | (1, 8) | 512 | integral even 2-elementary root lattice hyperbolic isotropic |
| 028R | \(\langle 2 \rangle \oplus A_{1}(-1)^{9}\) | 10 | (1, 9) | -1024 | integral even 2-elementary root lattice hyperbolic isotropic |
| 029G | \(\mathrm{Kum}(-1)\) | 16 | (16, 0) | 64 | integral even 2-elementary not a root lattice |
| 029H | \(\mathrm{Nik}(-1)\) | 8 | (8, 0) | 64 | integral even 2-elementary not a root lattice |
| 029J | \(U \oplus U(2) \oplus E_{8}(-2)\) | 12 | (2, 10) | 1024 | integral even 2-elementary not a root lattice isotropic |
| 029L | \(U^{3} \oplus E_{8}(-2)\) | 14 | (3, 11) | -256 | integral even 2-elementary root lattice isotropic |
| 029W | \(U \oplus \langle 2 \rangle^{2} \oplus \langle -2 \rangle^{2}\) | 6 | (3, 3) | -16 | integral even 2-elementary root lattice isotropic |
| 02A9 | \(\langle -1 \rangle^{3} \oplus \langle 2 \rangle\) | 4 | (1, 3) | -2 | integral odd 2-elementary root lattice hyperbolic isotropic |
| 02AO | \(\langle 1 \rangle \oplus \langle 2 \rangle^{2}\) | 3 | (3, 0) | 4 | integral odd 2-elementary root lattice |