Matroid 4.8.820
| Label | 4.8.820 |
|---|---|
| Id | r4_n8_00000000000000000000000000000**0**000000000000000**0**00000**0**0000** |
| Rank | 4 |
| n | 8 |
A graph with this cycle matroid; edge labels are elements.
Basic invariants
Representability
| Characteristic set | characteristic 0 and all primes [0] |
|---|---|
| Realizable | yes |
| Realizable char0 | yes |
| Regular | yes |
| Binary | yes |
| Ternary | yes |
| Quaternary | yes |
| Graphic | yes |
Realization space
| Status | computed_realization_space |
|---|---|
| Dimension of realization space | 1 |
| Expected dimension over ℤ | not computed |
| Components of realization space | 1 |
| Free realization space | yes |
| Principal ideal | yes |
| Birational type | rational how determined: birational type method: rigid |
| Birational type components | show[
{
"dim": 0,
"free_rank": 0,
"torus_rank": 0,
"core_dim": 0,
"qbar_components": 1,
"type": "rational",
"reason": "rigid: unique realization over Q (integer matrix)"
}
] |
| Good basis | not computed |
| Good basis v2 | 2,4,6,7 |
Geometry
Other
| Char poly splits | false |
|---|---|
| Simplicial arrangement | no |
| Realization space: n qbar components | 1 |
Tutte polynomial
\(T = x_{0}^{4} x_{1} + 2 x_{0}^{3} x_{1}^{2} + x_{0}^{3} x_{1} + x_{0}^{2} x_{1}^{3} + 2 x_{0}^{2} x_{1}^{2} + x_{0}^{2} x_{1} + 2 x_{0} x_{1}^{3} + 2 x_{0} x_{1}^{2} + x_{1}^{4} + x_{1}^{3}\)
Realization space
| Ring | \(\mathbb{Z}\) |
|---|---|
| Defining ideal | \(\left(0\right)\) |
| Inequations | none |
| Realization matrix | \(\begin{pmatrix}0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1\end{pmatrix}\) |
Combinatorics
Computed on the fly from the id.
Bases (14)
{2,4,6,7} {3,4,6,7} {2,5,6,7} {3,5,6,7} {2,4,6,8} {3,4,6,8} {2,5,6,8} {3,5,6,8} {2,4,7,8} {3,4,7,8} {2,5,7,8} {3,5,7,8} {4,6,7,8} {5,6,7,8}Non-bases (56)
{1,2,3,4} {1,2,3,5} {1,2,4,5} {1,3,4,5} {2,3,4,5} {1,2,3,6} {1,2,4,6} {1,3,4,6} {2,3,4,6} {1,2,5,6} {1,3,5,6} {2,3,5,6} {1,4,5,6} {2,4,5,6} {3,4,5,6} {1,2,3,7} {1,2,4,7} {1,3,4,7} {2,3,4,7} {1,2,5,7} {1,3,5,7} {2,3,5,7} {1,4,5,7} {2,4,5,7} {3,4,5,7} {1,2,6,7} {1,3,6,7} {2,3,6,7} {1,4,6,7} {1,5,6,7} {4,5,6,7} {1,2,3,8} {1,2,4,8} {1,3,4,8} {2,3,4,8} {1,2,5,8} {1,3,5,8} {2,3,5,8} {1,4,5,8} {2,4,5,8} {3,4,5,8} {1,2,6,8} {1,3,6,8} {2,3,6,8} {1,4,6,8} {1,5,6,8} {4,5,6,8} {1,2,7,8} {1,3,7,8} {2,3,7,8} {1,4,7,8} {1,5,7,8} {4,5,7,8} {1,6,7,8} {2,6,7,8} {3,6,7,8}Circuits (5)
{1} {2,3} {4,5} {2,6,7,8} {3,6,7,8}Flats by rank (24)
- rank 0 (1): {1}
- rank 1 (5): {1,2,3} {1,4,5} {1,6} {1,7} {1,8}
- rank 2 (10): {1,2,3,4,5} {1,2,3,6} {1,4,5,6} {1,2,3,7} {1,4,5,7} {1,6,7} {1,2,3,8} {1,4,5,8} {1,6,8} {1,7,8}
- rank 3 (7): {1,2,3,4,5,6} {1,2,3,4,5,7} {1,4,5,6,7} {1,2,3,4,5,8} {1,4,5,6,8} {1,4,5,7,8} {1,2,3,6,7,8}
- rank 4 (1): {1,2,3,4,5,6,7,8}
Hyperplanes (7)
{1,2,3,4,5,6} {1,2,3,4,5,7} {1,4,5,6,7} {1,2,3,4,5,8} {1,4,5,6,8} {1,4,5,7,8} {1,2,3,6,7,8}Dual (rank 4)
Revlex encoding in this labeling (not canonicalized, so not linked): **0000**0**00000**0**000000000000000**0**00000000000000000000000000000
Bases: {1,3,5,8} {1,2,5,8} {1,3,4,8} {1,2,4,8} {1,3,5,7} {1,2,5,7} {1,3,4,7} {1,2,4,7} {1,3,5,6} {1,2,5,6} {1,3,4,6} {1,2,4,6} {1,2,3,5} {1,2,3,4}
Loops: 1. Parallel classes of size > 1: {2,3} {4,5}.
Downloads
- Record as JSON
- OSCAR:
matroid_from_revlex_basis_encoding("00000000000000000000000000000**0**000000000000000**0**00000**0**0000**", 4, 8)