Collection
Root lattices
Lattices \(L\) with \(L = \mathbb{Z}\Phi(L)\).
A lattice \(L\) is a root lattice when \(L = \mathbb{Z}\Phi(L)\); the root sublattice \(\mathbb{Z}\Phi(L)\) is defined on the roots page.
ADE lattices
The lattices of the family irreducible-root-lattice
give \(A_n\), \(D_n\), \(E_6\), \(E_7\) and \(E_8\) in the basis of simple
roots. The dual
lattices of these lattices are separate lattices of the
catalogue.
292 lattices
This collection in the database
| Tag | Lattice | Rank | Signature | Determinant | Properties |
|---|---|---|---|---|---|
| 0001 | \(\langle 1 \rangle\) | 1 | (1, 0) | 1 | integral odd unimodular root lattice |
| 0002 | \(\langle 2 \rangle\) | 1 | (1, 0) | 2 | integral even 2-elementary root lattice |
| 0006 | \(\langle \frac{1}{2} \rangle\) | 1 | (1, 0) | 1/2 | not integral root lattice |
| 0011 | \(\mathrm{I}_{2,0}\) | 2 | (2, 0) | 1 | integral odd unimodular root lattice |
| 0012 | \(A_{2}\) | 2 | (2, 0) | 3 | integral even 3-elementary root lattice |
| 0014 | \(\langle 2 \rangle \oplus \langle -2 \rangle\) | 2 | (1, 1) | -4 | integral even 2-elementary root lattice hyperbolic isotropic |
| 0015 | \(\mathrm{I}_{1,1}\) | 2 | (1, 1) | -1 | integral odd unimodular root lattice hyperbolic isotropic |
| 0017 | \(A_{2}^{*}\) | 2 | (2, 0) | 1/3 | not integral root lattice |
| 0021 | \(\langle 1 \rangle \oplus \langle -2 \rangle\) | 2 | (1, 1) | -2 | integral odd 2-elementary root lattice hyperbolic anisotropic |
| 0022 | \(\mathrm{I}_{3,0}\) | 3 | (3, 0) | 1 | integral odd unimodular root lattice |
| 0023 | \(A_{3}\) | 3 | (3, 0) | 4 | integral even root lattice |
| 0024 | \(U(2) \oplus \langle -2 \rangle\) | 3 | (1, 2) | 8 | integral even 2-elementary root lattice hyperbolic isotropic |
| 0025 | \(\mathrm{I}_{1,2}\) | 3 | (1, 2) | 1 | integral odd unimodular root lattice hyperbolic isotropic |
| 0026 | \(U \oplus A_{1}(-1)\) | 3 | (1, 2) | 2 | integral even 2-elementary root lattice hyperbolic isotropic |
| 0029 | \(A_{2} \oplus A_{1}\) | 3 | (3, 0) | 6 | integral even root lattice |
| 0031 | \(\widetilde{A}_{2}\) | 3 | (2, 0) | 0 | integral even root lattice degenerate |
| 0033 | \(\mathrm{I}_{4,0}\) | 4 | (4, 0) | 1 | integral odd unimodular root lattice |
| 0034 | \(A_{4}\) | 4 | (4, 0) | 5 | integral even 5-elementary root lattice |
| 0035 | \(D_{4}\) | 4 | (4, 0) | 4 | integral even 2-elementary root lattice |
| 0038 | \(\mathrm{I}_{1,3}\) | 4 | (1, 3) | -1 | integral odd unimodular root lattice hyperbolic isotropic |
| 0039 | \(U \oplus A_{2}(-1)\) | 4 | (1, 3) | -3 | integral even 3-elementary root lattice hyperbolic isotropic |
| 0040 | \(U^{2}\) | 4 | (2, 2) | 1 | integral even unimodular root lattice isotropic |
| 0042 | \(D_{4}^{*}\) | 4 | (4, 0) | 1/4 | not integral root lattice |
| 0043 | \(A_{2}^{2}\) | 4 | (4, 0) | 9 | integral even 3-elementary root lattice |
| 0046 | \(\mathrm{I}_{2,2}\) | 4 | (2, 2) | 1 | integral odd unimodular root lattice isotropic |
| 0047 | \(\widetilde{A}_{3}\) | 4 | (3, 0) | 0 | integral even root lattice degenerate |
| 0049 | \(\mathrm{I}_{5,0}\) | 5 | (5, 0) | 1 | integral odd unimodular root lattice |
| 0050 | \(A_{5}\) | 5 | (5, 0) | 6 | integral even root lattice |
| 0051 | \(D_{5}\) | 5 | (5, 0) | 4 | integral even root lattice |
| 0052 | \(\mathrm{I}_{1,4}\) | 5 | (1, 4) | 1 | integral odd unimodular root lattice hyperbolic isotropic |
| 0053 | \(U \oplus A_{3}(-1)\) | 5 | (1, 4) | 4 | integral even root lattice hyperbolic isotropic |
| 0056 | \(A_{3} \oplus A_{2}\) | 5 | (5, 0) | 12 | integral even root lattice |
| 0057 | \(D_{4} \oplus A_{1}\) | 5 | (5, 0) | 8 | integral even 2-elementary root lattice |
| 0059 | \(\mathrm{I}_{2,3}\) | 5 | (2, 3) | -1 | integral odd unimodular root lattice isotropic |
| 0060 | \(\widetilde{D}_{4}\) | 5 | (4, 0) | 0 | integral even root lattice degenerate |
| 0061 | \(\mathrm{I}_{6,0}\) | 6 | (6, 0) | 1 | integral odd unimodular root lattice |
| 0062 | \(A_{6}\) | 6 | (6, 0) | 7 | integral even 7-elementary root lattice |
| 0063 | \(D_{6}\) | 6 | (6, 0) | 4 | integral even 2-elementary root lattice |
| 0064 | \(E_{6}\) | 6 | (6, 0) | 3 | integral even 3-elementary root lattice |
| 0065 | \(\mathrm{I}_{1,5}\) | 6 | (1, 5) | -1 | integral odd unimodular root lattice hyperbolic isotropic |
| 0066 | \(U \oplus D_{4}(-1)\) | 6 | (1, 5) | -4 | integral even 2-elementary root lattice hyperbolic isotropic |
| 0067 | \(U^{3}\) | 6 | (3, 3) | -1 | integral even unimodular root lattice isotropic |
| 0071 | \(A_{2}^{3}\) | 6 | (6, 0) | 27 | integral even 3-elementary root lattice |
| 0072 | \(A_{3}^{2}\) | 6 | (6, 0) | 16 | integral even root lattice |
| 0074 | \(\mathrm{I}_{2,4}\) | 6 | (2, 4) | 1 | integral odd unimodular root lattice isotropic |
| 0075 | \(\widetilde{D}_{5}\) | 6 | (5, 0) | 0 | integral even root lattice degenerate |
| 0076 | \(\mathrm{I}_{7,0}\) | 7 | (7, 0) | 1 | integral odd unimodular root lattice |
| 0077 | \(A_{7}\) | 7 | (7, 0) | 8 | integral even root lattice |
| 0078 | \(D_{7}\) | 7 | (7, 0) | 4 | integral even root lattice |
| 0079 | \(E_{7}\) | 7 | (7, 0) | 2 | integral even 2-elementary root lattice |
| 0081 | \(\mathrm{I}_{1,6}\) | 7 | (1, 6) | 1 | integral odd unimodular root lattice hyperbolic isotropic |
| 0082 | \(U \oplus D_{5}(-1)\) | 7 | (1, 6) | 4 | integral even root lattice hyperbolic isotropic |
| 0086 | \(D_{4} \oplus A_{3}\) | 7 | (7, 0) | 16 | integral even root lattice |
| 0087 | \(E_{6} \oplus A_{1}\) | 7 | (7, 0) | 6 | integral even root lattice |
| 0089 | \(\mathrm{I}_{2,5}\) | 7 | (2, 5) | -1 | integral odd unimodular root lattice isotropic |
| 0090 | \(\widetilde{E}_{6}\) | 7 | (6, 0) | 0 | integral even root lattice degenerate |
| 0091 | \(\mathrm{I}_{8,0}\) | 8 | (8, 0) | 1 | integral odd unimodular root lattice |
| 0092 | \(A_{8}\) | 8 | (8, 0) | 9 | integral even root lattice |
| 0093 | \(D_{8}\) | 8 | (8, 0) | 4 | integral even 2-elementary root lattice |
| 0094 | \(E_{8}\) | 8 | (8, 0) | 1 | integral even unimodular root lattice |
| 0097 | \(\mathrm{I}_{1,7}\) | 8 | (1, 7) | -1 | integral odd unimodular root lattice hyperbolic isotropic |
| 0098 | \(U \oplus E_{6}(-1)\) | 8 | (1, 7) | -3 | integral even 3-elementary root lattice hyperbolic isotropic |
| 0099 | \(U^{4}\) | 8 | (4, 4) | 1 | integral even unimodular root lattice isotropic |
| 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 |
| 0105 | \(\mathrm{I}_{2,6}\) | 8 | (2, 6) | 1 | integral odd unimodular root lattice isotropic |
| 0106 | \(\widetilde{E}_{7}\) | 8 | (7, 0) | 0 | integral even root lattice degenerate |
| 0107 | \(\mathrm{I}_{9,0}\) | 9 | (9, 0) | 1 | integral odd unimodular root lattice |
| 0108 | \(A_{9}\) | 9 | (9, 0) | 10 | integral even root lattice |
| 0109 | \(D_{9}\) | 9 | (9, 0) | 4 | integral even root lattice |
| 0112 | \(\mathrm{I}_{1,8}\) | 9 | (1, 8) | 1 | integral odd unimodular root lattice hyperbolic isotropic |
| 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 |
| 0118 | \(\mathrm{I}_{2,7}\) | 9 | (2, 7) | -1 | integral odd unimodular root lattice isotropic |
| 0119 | \(\widetilde{E}_{8}\) | 9 | (8, 0) | 0 | integral even root lattice degenerate |
| 0120 | \(\mathrm{I}_{10,0}\) | 10 | (10, 0) | 1 | integral odd unimodular root lattice |
| 0121 | \(A_{10}\) | 10 | (10, 0) | 11 | integral even 11-elementary root lattice |
| 0122 | \(D_{10}\) | 10 | (10, 0) | 4 | integral even 2-elementary root lattice |
| 0125 | \(U(2) \oplus E_{8}(-2)\) | 10 | (1, 9) | -1024 | integral even 2-elementary root lattice hyperbolic isotropic |
| 0126 | \(\mathrm{I}_{1,9}\) | 10 | (1, 9) | -1 | integral odd unimodular root lattice hyperbolic isotropic |
| 0127 | \(U \oplus D_{8}(-1)\) | 10 | (1, 9) | -4 | integral even 2-elementary root lattice hyperbolic isotropic |
| 0128 | \(U \oplus E_{8}(-1)\) | 10 | (1, 9) | -1 | integral even unimodular root lattice hyperbolic isotropic |
| 0131 | \(E_{8} \oplus A_{2}\) | 10 | (10, 0) | 3 | integral even 3-elementary root lattice |
| 0133 | \(\mathrm{I}_{2,8}\) | 10 | (2, 8) | 1 | integral odd unimodular root lattice isotropic |
| 0134 | \(\mathrm{I}_{11,0}\) | 11 | (11, 0) | 1 | integral odd unimodular root lattice |
| 0135 | \(A_{11}\) | 11 | (11, 0) | 12 | integral even root lattice |
| 0136 | \(D_{11}\) | 11 | (11, 0) | 4 | integral even root lattice |
| 0138 | \(\langle 2 \rangle \oplus A_{1}(-1)^{10}\) | 11 | (1, 10) | 2048 | integral even 2-elementary root lattice hyperbolic isotropic reflective |
| 0139 | \(\mathrm{I}_{1,10}\) | 11 | (1, 10) | 1 | integral odd unimodular root lattice hyperbolic isotropic |
| 0140 | \(U \oplus D_{9}(-1)\) | 11 | (1, 10) | 4 | integral even root lattice hyperbolic isotropic |
| 0143 | \(E_{8} \oplus A_{3}\) | 11 | (11, 0) | 4 | integral even root lattice |
| 0145 | \(\mathrm{I}_{2,9}\) | 11 | (2, 9) | -1 | integral odd unimodular root lattice isotropic |
| 0146 | \(\mathrm{I}_{12,0}\) | 12 | (12, 0) | 1 | integral odd unimodular root lattice |
| 0147 | \(A_{12}\) | 12 | (12, 0) | 13 | integral even 13-elementary root lattice |
| 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 |
| 0154 | \(\mathrm{I}_{1,11}\) | 12 | (1, 11) | -1 | integral odd unimodular root lattice hyperbolic isotropic |
| 0155 | \(U \oplus D_{10}(-1)\) | 12 | (1, 11) | -4 | integral even 2-elementary root lattice hyperbolic isotropic |
| 0156 | \(U^{2} \oplus E_{8}(-1)\) | 12 | (2, 10) | 1 | integral even unimodular root lattice isotropic |
| 0159 | \(E_{8} \oplus D_{4}\) | 12 | (12, 0) | 4 | integral even 2-elementary root lattice |
| 0160 | \(E_{6}^{2}\) | 12 | (12, 0) | 9 | integral even 3-elementary root lattice |
| 0162 | \(\mathrm{I}_{2,10}\) | 12 | (2, 10) | 1 | integral odd unimodular root lattice isotropic |
| 0163 | \(\mathrm{I}_{13,0}\) | 13 | (13, 0) | 1 | integral odd unimodular root lattice |
| 0164 | \(A_{13}\) | 13 | (13, 0) | 14 | integral even root lattice |
| 0165 | \(D_{13}\) | 13 | (13, 0) | 4 | integral even 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 |
| 0168 | \(\mathrm{I}_{1,12}\) | 13 | (1, 12) | 1 | integral odd unimodular root lattice hyperbolic isotropic |
| 0169 | \(U \oplus D_{11}(-1)\) | 13 | (1, 12) | 4 | integral even root lattice hyperbolic isotropic |
| 0172 | \(E_{8} \oplus D_{5}\) | 13 | (13, 0) | 4 | integral even root lattice |
| 0174 | \(\mathrm{I}_{2,11}\) | 13 | (2, 11) | -1 | integral odd unimodular root lattice isotropic |
| 0175 | \(\mathrm{I}_{14,0}\) | 14 | (14, 0) | 1 | integral odd unimodular root lattice |
| 0176 | \(A_{14}\) | 14 | (14, 0) | 15 | integral even root lattice |
| 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 |
| 0181 | \(\mathrm{I}_{1,13}\) | 14 | (1, 13) | -1 | integral odd unimodular root lattice hyperbolic isotropic |
| 0182 | \(U \oplus D_{12}(-1)\) | 14 | (1, 13) | -4 | integral even 2-elementary root lattice hyperbolic isotropic |
| 0183 | \(U^{3} \oplus E_{8}(-1)\) | 14 | (3, 11) | -1 | integral even unimodular root lattice isotropic |
| 0186 | \(E_{8} \oplus E_{6}\) | 14 | (14, 0) | 3 | integral even 3-elementary root lattice |
| 0187 | \(E_{7}^{2}\) | 14 | (14, 0) | 4 | integral even 2-elementary root lattice |
| 0189 | \(\mathrm{I}_{2,12}\) | 14 | (2, 12) | 1 | integral odd unimodular root lattice isotropic |
| 0190 | \(\mathrm{I}_{15,0}\) | 15 | (15, 0) | 1 | integral odd unimodular root lattice |
| 0191 | \(A_{15}\) | 15 | (15, 0) | 16 | integral even root lattice |
| 0192 | \(D_{15}\) | 15 | (15, 0) | 4 | integral even 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 |
| 0195 | \(\mathrm{I}_{1,14}\) | 15 | (1, 14) | 1 | integral odd unimodular root lattice hyperbolic isotropic |
| 0196 | \(U \oplus D_{13}(-1)\) | 15 | (1, 14) | 4 | integral even root lattice hyperbolic isotropic |
| 0199 | \(E_{8} \oplus E_{7}\) | 15 | (15, 0) | 2 | integral even 2-elementary root lattice |
| 0201 | \(\mathrm{I}_{2,13}\) | 15 | (2, 13) | -1 | integral odd unimodular root lattice isotropic |
| 0202 | \(\mathrm{I}_{16,0}\) | 16 | (16, 0) | 1 | integral odd unimodular root lattice |
| 0203 | \(A_{16}\) | 16 | (16, 0) | 17 | integral even 17-elementary root lattice |
| 0204 | \(D_{16}\) | 16 | (16, 0) | 4 | integral even 2-elementary 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 |
| 0208 | \(\mathrm{I}_{1,15}\) | 16 | (1, 15) | -1 | integral odd unimodular root lattice hyperbolic isotropic |
| 0209 | \(U \oplus D_{14}(-1)\) | 16 | (1, 15) | -4 | integral even 2-elementary root lattice hyperbolic isotropic |
| 0210 | \(U^{4} \oplus E_{8}(-1)\) | 16 | (4, 12) | 1 | integral even unimodular root lattice isotropic |
| 0213 | \(E_{8}^{2}\) | 16 | (16, 0) | 1 | integral even unimodular root lattice |
| 0215 | \(\mathrm{I}_{2,14}\) | 16 | (2, 14) | 1 | integral odd unimodular root lattice isotropic |
| 0216 | \(\mathrm{I}_{17,0}\) | 17 | (17, 0) | 1 | integral odd unimodular root lattice |
| 0217 | \(A_{17}\) | 17 | (17, 0) | 18 | integral even root lattice |
| 0218 | \(D_{17}\) | 17 | (17, 0) | 4 | integral even root lattice |
| 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 |
| 0221 | \(\mathrm{I}_{1,16}\) | 17 | (1, 16) | 1 | integral odd unimodular root lattice hyperbolic isotropic |
| 0222 | \(U \oplus D_{15}(-1)\) | 17 | (1, 16) | 4 | integral even root lattice hyperbolic isotropic |
| 0225 | \(E_{8}^{2} \oplus A_{1}\) | 17 | (17, 0) | 2 | integral even 2-elementary root lattice |
| 0227 | \(\mathrm{I}_{2,15}\) | 17 | (2, 15) | -1 | integral odd unimodular root lattice isotropic |
| 0228 | \(\mathrm{I}_{18,0}\) | 18 | (18, 0) | 1 | integral odd unimodular root lattice |
| 0229 | \(A_{18}\) | 18 | (18, 0) | 19 | integral even 19-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 |
| 0234 | \(\mathrm{I}_{1,17}\) | 18 | (1, 17) | -1 | integral odd unimodular root lattice hyperbolic isotropic |
| 0235 | \(U \oplus D_{16}(-1)\) | 18 | (1, 17) | -4 | integral even 2-elementary root lattice hyperbolic isotropic |
| 0236 | \(U \oplus E_{8}(-1)^{2}\) | 18 | (1, 17) | -1 | integral even unimodular root lattice hyperbolic isotropic |
| 0239 | \(E_{8}^{2} \oplus A_{2}\) | 18 | (18, 0) | 3 | integral even 3-elementary root lattice |
| 0241 | \(\mathrm{I}_{2,16}\) | 18 | (2, 16) | 1 | integral odd unimodular root lattice isotropic |
| 0242 | \(\mathrm{I}_{19,0}\) | 19 | (19, 0) | 1 | integral odd unimodular root lattice |
| 0243 | \(A_{19}\) | 19 | (19, 0) | 20 | integral even root lattice |
| 0244 | \(D_{19}\) | 19 | (19, 0) | 4 | integral even root lattice |
| 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 |
| 0247 | \(\mathrm{I}_{1,18}\) | 19 | (1, 18) | 1 | integral odd unimodular root lattice hyperbolic isotropic |
| 0248 | \(U \oplus D_{17}(-1)\) | 19 | (1, 18) | 4 | integral even root lattice hyperbolic isotropic |
| 0251 | \(E_{8}^{2} \oplus A_{3}\) | 19 | (19, 0) | 4 | integral even root lattice |
| 0253 | \(\mathrm{I}_{2,17}\) | 19 | (2, 17) | -1 | integral odd unimodular root lattice isotropic |
| 0254 | \(\mathrm{I}_{20,0}\) | 20 | (20, 0) | 1 | integral odd unimodular root lattice |
| 0255 | \(A_{20}\) | 20 | (20, 0) | 21 | integral even root lattice |
| 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 |
| 0260 | \(\mathrm{I}_{1,19}\) | 20 | (1, 19) | -1 | integral odd unimodular root lattice hyperbolic isotropic |
| 0261 | \(U^{2} \oplus E_{8}(-1)^{2}\) | 20 | (2, 18) | 1 | integral even unimodular root lattice isotropic |
| 0264 | \(E_{8}^{2} \oplus D_{4}\) | 20 | (20, 0) | 4 | integral even 2-elementary root lattice |
| 0266 | \(\mathrm{I}_{2,18}\) | 20 | (2, 18) | 1 | integral odd unimodular root lattice isotropic |
| 0268 | \(\langle -1 \rangle^{2} \oplus \langle 3 \rangle\) | 3 | (1, 2) | 3 | integral odd 3-elementary root lattice hyperbolic anisotropic |
| 0269 | \(\langle -1 \rangle^{3} \oplus \langle 7 \rangle\) | 4 | (1, 3) | -7 | integral odd 7-elementary root lattice hyperbolic anisotropic |
| 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 |
| 027E | \(U^{3} \oplus E_{8}(-1)^{2}\) | 22 | (3, 19) | -1 | integral even unimodular root lattice isotropic |
| 027F | \(U^{4} \oplus E_{8}(-1)^{2}\) | 24 | (4, 20) | 1 | integral even unimodular root lattice isotropic |
| 027G | \(U \oplus E_{8}(-1)^{3}\) | 26 | (1, 25) | -1 | integral even unimodular root lattice hyperbolic isotropic |
| 027H | \(\mathrm{I}_{1,20}\) | 21 | (1, 20) | 1 | integral odd unimodular root lattice hyperbolic isotropic |
| 027I | \(\mathrm{I}_{1,21}\) | 22 | (1, 21) | -1 | integral odd unimodular root lattice hyperbolic isotropic |
| 027J | \(\mathrm{I}_{1,22}\) | 23 | (1, 22) | 1 | integral odd unimodular root lattice hyperbolic isotropic |
| 027K | \(\mathrm{I}_{1,23}\) | 24 | (1, 23) | -1 | integral odd unimodular root lattice hyperbolic isotropic |
| 027L | \(\mathrm{I}_{1,24}\) | 25 | (1, 24) | 1 | integral odd unimodular root lattice hyperbolic isotropic |
| 027M | \(\mathrm{I}_{1,25}\) | 26 | (1, 25) | -1 | integral odd unimodular root lattice hyperbolic isotropic |
| 027N | \(\mathrm{I}_{2,19}\) | 21 | (2, 19) | -1 | integral odd unimodular root lattice isotropic |
| 027O | \(\mathrm{I}_{2,20}\) | 22 | (2, 20) | 1 | integral odd unimodular root lattice isotropic |
| 027P | \(\mathrm{I}_{2,21}\) | 23 | (2, 21) | -1 | integral odd unimodular root lattice isotropic |
| 027Q | \(\mathrm{I}_{2,22}\) | 24 | (2, 22) | 1 | integral odd unimodular root lattice isotropic |
| 027R | \(\mathrm{I}_{2,23}\) | 25 | (2, 23) | -1 | integral odd unimodular root lattice isotropic |
| 027S | \(\mathrm{I}_{2,24}\) | 26 | (2, 24) | 1 | integral odd unimodular root lattice isotropic |
| 027T | \(\mathrm{I}_{21,0}\) | 21 | (21, 0) | 1 | integral odd unimodular root lattice |
| 027U | \(\mathrm{I}_{22,0}\) | 22 | (22, 0) | 1 | integral odd unimodular root lattice |
| 027V | \(\mathrm{I}_{23,0}\) | 23 | (23, 0) | 1 | integral odd unimodular root lattice |
| 027W | \(\mathrm{I}_{24,0}\) | 24 | (24, 0) | 1 | integral odd unimodular root lattice |
| 027X | \(\mathrm{I}_{25,0}\) | 25 | (25, 0) | 1 | integral odd unimodular root lattice |
| 027Y | \(\mathrm{I}_{26,0}\) | 26 | (26, 0) | 1 | integral odd unimodular root lattice |
| 027Z | \(U^{3} \oplus E_{8}(-1)^{2} \oplus \langle -2 \rangle\) | 23 | (3, 20) | 2 | integral even 2-elementary root lattice isotropic |
| 0280 | \(U^{3} \oplus E_{8}(-1)^{2} \oplus \langle -4 \rangle\) | 23 | (3, 20) | 4 | integral even root lattice isotropic |
| 0281 | \(U^{3} \oplus E_{8}(-1)^{2} \oplus \langle -6 \rangle\) | 23 | (3, 20) | 6 | integral even root lattice isotropic |
| 0282 | \(U^{3} \oplus E_{8}(-1)^{2} \oplus \langle -8 \rangle\) | 23 | (3, 20) | 8 | integral even root lattice isotropic |
| 0283 | \(U^{3} \oplus \langle -6 \rangle\) | 7 | (3, 4) | 6 | integral even root lattice isotropic |
| 0284 | \(U^{3} \oplus \langle -8 \rangle\) | 7 | (3, 4) | 8 | integral even root lattice isotropic |
| 0285 | \(U^{3} \oplus \langle -10 \rangle\) | 7 | (3, 4) | 10 | integral even root lattice isotropic |
| 0286 | \(U^{3} \oplus \langle -12 \rangle\) | 7 | (3, 4) | 12 | integral even root lattice isotropic |
| 0287 | \(U^{3} \oplus \langle -2 \rangle^{2}\) | 8 | (3, 5) | -4 | integral even 2-elementary root lattice isotropic |
| 0288 | \(U^{3} \oplus E_{8}(-1)^{2} \oplus A_{2}(-1)\) | 24 | (3, 21) | -3 | integral even 3-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 |
| 028A | \(U^{2} \oplus E_{8}(-1)^{2} \oplus \langle -4 \rangle\) | 21 | (2, 19) | -4 | integral even root lattice isotropic |
| 028B | \(U^{2} \oplus E_{8}(-1)^{2} \oplus \langle -6 \rangle\) | 21 | (2, 19) | -6 | integral even root lattice isotropic |
| 028C | \(U^{2} \oplus E_{8}(-1)^{2} \oplus \langle -8 \rangle\) | 21 | (2, 19) | -8 | integral even root lattice isotropic |
| 028D | \(U^{2} \oplus E_{8}(-1)^{2} \oplus \langle -10 \rangle\) | 21 | (2, 19) | -10 | integral even root lattice isotropic |
| 028E | \(U^{2} \oplus E_{8}(-1)^{2} \oplus \langle -12 \rangle\) | 21 | (2, 19) | -12 | integral even root lattice isotropic |
| 028F | \(U^{2} \oplus E_{8}(-1)^{2} \oplus \langle -14 \rangle\) | 21 | (2, 19) | -14 | integral even root lattice isotropic |
| 028G | \(U^{2} \oplus E_{8}(-1)^{2} \oplus \langle -16 \rangle\) | 21 | (2, 19) | -16 | integral even root lattice isotropic |
| 028H | \(U^{2} \oplus E_{8}(-1)^{2} \oplus \langle -18 \rangle\) | 21 | (2, 19) | -18 | integral even root lattice isotropic |
| 028I | \(U^{2} \oplus E_{8}(-1)^{2} \oplus \langle -20 \rangle\) | 21 | (2, 19) | -20 | integral even root lattice isotropic |
| 028J | \(U^{2} \oplus E_{8}(-1)^{2} \oplus \langle -22 \rangle\) | 21 | (2, 19) | -22 | integral even root lattice isotropic |
| 028K | \(U^{2} \oplus E_{8}(-1)^{2} \oplus \langle -24 \rangle\) | 21 | (2, 19) | -24 | integral even 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 |
| 0294 | \(E_{8}^{3}\) | 24 | (24, 0) | 1 | integral even unimodular root lattice |
| 029I | \(U^{2} \oplus E_{8}(-1)^{2} \oplus A_{2}(-1)\) | 22 | (2, 20) | 3 | integral even 3-elementary root lattice isotropic |
| 029K | \(U^{2} \oplus E_{8}(-1)^{3}\) | 28 | (2, 26) | 1 | integral even unimodular root lattice isotropic |
| 029L | \(U^{3} \oplus E_{8}(-2)\) | 14 | (3, 11) | -256 | integral even 2-elementary root lattice isotropic |
| 029M | \(U \oplus U(3)^{2} \oplus A_{2}(-1)^{2}\) | 10 | (3, 7) | -729 | integral even 3-elementary root lattice isotropic |
| 029O | \(U \oplus U(4)^{2} \oplus \langle -2 \rangle^{2}\) | 8 | (3, 5) | -1024 | integral even root lattice isotropic |
| 029P | \(U(3) \oplus A_{2}(2) \oplus A_{2}(-1)^{2}\) | 8 | (3, 5) | -972 | integral even root lattice isotropic |
| 029R | \(\begin{pmatrix} -3 & -1 & -1 \\ -1 & -3 & 1 \\ -1 & 1 & -3 \end{pmatrix} \oplus \langle 1 \rangle^{2}\) | 5 | (2, 3) | -16 | integral odd root lattice isotropic |
| 029S | \(U \oplus U(5)^{2}\) | 6 | (3, 3) | -625 | integral even 5-elementary root lattice isotropic |
| 029V | \(U^{3} \oplus \langle -4 \rangle\) | 7 | (3, 4) | 4 | integral even 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 |
| 029X | \(U(4) \oplus \langle -1 \rangle \oplus \langle -2 \rangle\) | 4 | (1, 3) | -32 | integral odd root lattice hyperbolic isotropic |
| 029Y | \(U(3)^{2} \oplus \begin{pmatrix} 2 & 3 \\ 3 & 0 \end{pmatrix}\) | 6 | (3, 3) | -729 | integral even root lattice isotropic |
| 029Z | \(U(5) \oplus \begin{pmatrix} -4 & -2 \\ -2 & -6 \end{pmatrix}\) | 4 | (1, 3) | -500 | integral even root lattice hyperbolic isotropic |
| 02A0 | \(U(7) \oplus \begin{pmatrix} -2 & -1 \\ -1 & -4 \end{pmatrix}\) | 4 | (1, 3) | -343 | integral even 7-elementary root lattice hyperbolic isotropic |
| 02A2 | \(U \oplus \langle -2 \rangle \oplus \langle 2 \rangle \oplus \langle -4 \rangle\) | 5 | (2, 3) | -16 | integral even root lattice isotropic |
| 02A3 | \(A_{2}(-1) \oplus \langle -6 \rangle \oplus \langle 18 \rangle\) | 4 | (1, 3) | -324 | integral even root lattice hyperbolic isotropic |
| 02A5 | \(U \oplus A_{2}(-1) \oplus \langle 4 \rangle\) | 5 | (2, 3) | -12 | integral even root lattice isotropic |
| 02A6 | \(U \oplus \langle -6 \rangle^{2}\) | 4 | (1, 3) | -36 | integral even root lattice hyperbolic isotropic |
| 02A7 | \(\langle 1 \rangle \oplus A_{2}(4)\) | 3 | (3, 0) | 48 | integral odd root lattice |
| 02A9 | \(\langle -1 \rangle^{3} \oplus \langle 2 \rangle\) | 4 | (1, 3) | -2 | integral odd 2-elementary root lattice hyperbolic isotropic |
| 02AC | \(\langle 1 \rangle \oplus A_{2}(3)\) | 3 | (3, 0) | 27 | integral odd root lattice |
| 02AD | \(A_{2}(-1) \oplus \langle -2 \rangle \oplus \langle 4 \rangle\) | 4 | (1, 3) | -24 | integral even root lattice hyperbolic isotropic |
| 02AI | \(\begin{pmatrix} -2 & 0 & -1 & 0 \\ 0 & -2 & -1 & 0 \\ -1 & -1 & -2 & -2 \\ 0 & 0 & -2 & 0 \end{pmatrix}\) | 4 | (1, 3) | -16 | integral even root lattice hyperbolic isotropic |
| 02AJ | \(\langle 1 \rangle \oplus \langle 2 \rangle \oplus \langle 3 \rangle\) | 3 | (3, 0) | 6 | integral odd root lattice |
| 02AK | \(\langle 1 \rangle \oplus A_{2}\) | 3 | (3, 0) | 3 | integral odd 3-elementary root lattice |
| 02AO | \(\langle 1 \rangle \oplus \langle 2 \rangle^{2}\) | 3 | (3, 0) | 4 | integral odd 2-elementary root lattice |