Collection
Hyperbolic lattices
Nondegenerate lattices of rank at least 2 with signature \((1, n)\) or \((n, 1)\).
A lattice of rank \(n + 1 \geq 2\) is hyperbolic when it is nondegenerate of signature \((1, n)\) or \((n, 1)\). The page of each member states whether it is reflective, when this is known.
124 lattices
This collection in the database
| Tag | Lattice | Rank | Signature | Determinant | Properties |
|---|---|---|---|---|---|
| 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 |
| 0015 | \(\mathrm{I}_{1,1}\) | 2 | (1, 1) | -1 | integral odd unimodular root lattice hyperbolic isotropic |
| 0016 | \(U\) | 2 | (1, 1) | -1 | integral even unimodular not a 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 0152 | \(U \oplus A_{1}(-1)^{10}\) | 12 | (1, 11) | -1024 | integral even 2-elementary root lattice hyperbolic isotropic reflective |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 029X | \(U(4) \oplus \langle -1 \rangle \oplus \langle -2 \rangle\) | 4 | (1, 3) | -32 | integral odd root lattice hyperbolic 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 |
| 02A3 | \(A_{2}(-1) \oplus \langle -6 \rangle \oplus \langle 18 \rangle\) | 4 | (1, 3) | -324 | integral even root lattice hyperbolic isotropic |
| 02A4 | \(U \oplus A_{2}(-2)\) | 4 | (1, 3) | -12 | integral even not a root lattice hyperbolic isotropic |
| 02A6 | \(U \oplus \langle -6 \rangle^{2}\) | 4 | (1, 3) | -36 | integral even root lattice hyperbolic isotropic |
| 02A8 | \(U \oplus A_{2}(-10)\) | 4 | (1, 3) | -300 | integral even not a root lattice hyperbolic isotropic |
| 02A9 | \(\langle -1 \rangle^{3} \oplus \langle 2 \rangle\) | 4 | (1, 3) | -2 | integral odd 2-elementary root lattice hyperbolic isotropic |
| 02AA | \(U(3) \oplus A_{2}(-4)\) | 4 | (1, 3) | -432 | integral even not a root lattice hyperbolic isotropic |
| 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 |