Matroid 4.8.228
| Label | 4.8.228 |
|---|---|
| Id | r4_n8_0********0****0*****0**0*****************0****0***********0*******0*** |
| Rank | 4 |
| n | 8 |
Basic invariants
Representability
| Characteristic set | characteristic 0 and all primes [0] |
|---|---|
| Realizable | yes |
| Realizable char0 | yes |
| Regular | no |
| Binary | no |
| Ternary | no |
| Quaternary | yes |
| Graphic | no |
Realization space
| Status | computed_realization_space |
|---|---|
| Dimension of realization space | 2 |
| Expected dimension over ℤ | not computed |
| Components of realization space | 2 |
| Free realization space | no |
| Principal ideal | yes |
| Birational type | point over Q(a)/(a^2 - a - 1) how determined: birational type method: point_field |
| Birational type components | show[
{
"dim": 0,
"free_rank": 0,
"torus_rank": 0,
"core_dim": 0,
"qbar_components": 2,
"type": "point over Q(a)/(a^2 - a - 1)",
"field": "Q(a)/(a^2 - a - 1)",
"minpoly": "T^2 - T - 1",
"degree": 2,
"core_vars": [
"x1"
],
"core_gens": [
"x1^2 + x1 - 1"
]
}
] |
| Good basis | not computed |
| Good basis v2 | 1,2,3,8 |
Geometry
Other
| Char poly splits | false |
|---|---|
| Simplicial arrangement | no |
| Realization space: n qbar components | 2 |
Tutte polynomial
\(T = x_{0}^{4} + 4 x_{0}^{3} + 10 x_{0}^{2} + 9 x_{0} x_{1} + 11 x_{0} + x_{1}^{4} + 4 x_{1}^{3} + 10 x_{1}^{2} + 11 x_{1}\)
Realization space
| Ring | \(\mathbb{Z}[x_{1}]\) |
|---|---|
| Defining ideal | \(\left(x_{1}^{2} + x_{1} - 1\right)\) |
| Inequations | none |
| Realization matrix | \(\begin{pmatrix}1 & 0 & 0 & 1 & 0 & 1 & 1 & 0 \\ 0 & 1 & 0 & 1 & 1 & 0 & x_{1} & 0 \\ 0 & 0 & 1 & 1 & -x_{1} & x_{1} & x_{1} & 0 \\ 0 & 0 & 0 & 0 & 1 & -1 & x_{1} & 1\end{pmatrix}\) |
Combinatorics
Computed on the fly from the id.
Bases (61)
{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} {1,3,5,6} {2,3,5,6} {1,4,5,6} {2,4,5,6} {1,2,3,7} {1,2,4,7} {1,3,4,7} {2,3,4,7} {1,2,5,7} {2,3,5,7} {1,4,5,7} {3,4,5,7} {1,2,6,7} {1,3,6,7} {2,3,6,7} {1,4,6,7} {2,4,6,7} {3,4,6,7} {1,5,6,7} {2,5,6,7} {3,5,6,7} {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} {1,4,5,8} {2,4,5,8} {3,4,5,8} {1,2,6,8} {2,3,6,8} {1,4,6,8} {2,4,6,8} {3,4,6,8} {1,5,6,8} {2,5,6,8} {3,5,6,8} {4,5,6,8} {1,2,7,8} {1,3,7,8} {2,3,7,8} {2,4,7,8} {3,4,7,8} {1,5,7,8} {2,5,7,8} {3,5,7,8} {4,5,7,8} {1,6,7,8} {3,6,7,8} {4,6,7,8} {5,6,7,8}Non-bases (9)
{1,2,3,4} {1,2,5,6} {3,4,5,6} {1,3,5,7} {2,4,5,7} {2,3,5,8} {1,3,6,8} {1,4,7,8} {2,6,7,8}Circuits (29)
{1,2,3,4} {1,2,5,6} {3,4,5,6} {1,3,5,7} {2,4,5,7} {1,2,3,6,7} {1,2,4,6,7} {1,3,4,6,7} {2,3,4,6,7} {2,3,5,6,7} {1,4,5,6,7} {2,3,5,8} {1,2,4,5,8} {1,3,4,5,8} {1,3,6,8} {1,2,4,6,8} {2,3,4,6,8} {1,4,5,6,8} {2,4,5,6,8} {1,2,3,7,8} {1,4,7,8} {2,3,4,7,8} {1,2,5,7,8} {3,4,5,7,8} {2,6,7,8} {3,4,6,7,8} {1,5,6,7,8} {3,5,6,7,8} {4,5,6,7,8}Flats by rank (67)
- rank 0 (1): {}
- rank 1 (8): {1} {2} {3} {4} {5} {6} {7} {8}
- rank 2 (28): {1,2} {1,3} {2,3} {1,4} {2,4} {3,4} {1,5} {2,5} {3,5} {4,5} {1,6} {2,6} {3,6} {4,6} {5,6} {1,7} {2,7} {3,7} {4,7} {5,7} {6,7} {1,8} {2,8} {3,8} {4,8} {5,8} {6,8} {7,8}
- rank 3 (29): {1,2,3,4} {1,4,5} {2,3,6} {1,4,6} {2,4,6} {1,2,5,6} {3,4,5,6} {1,2,7} {2,3,7} {3,4,7} {1,3,5,7} {2,4,5,7} {1,6,7} {3,6,7} {4,6,7} {5,6,7} {1,2,8} {2,4,8} {3,4,8} {1,5,8} {2,3,5,8} {4,5,8} {1,3,6,8} {4,6,8} {5,6,8} {3,7,8} {1,4,7,8} {5,7,8} {2,6,7,8}
- rank 4 (1): {1,2,3,4,5,6,7,8}
Hyperplanes (29)
{1,2,3,4} {1,4,5} {2,3,6} {1,4,6} {2,4,6} {1,2,5,6} {3,4,5,6} {1,2,7} {2,3,7} {3,4,7} {1,3,5,7} {2,4,5,7} {1,6,7} {3,6,7} {4,6,7} {5,6,7} {1,2,8} {2,4,8} {3,4,8} {1,5,8} {2,3,5,8} {4,5,8} {1,3,6,8} {4,6,8} {5,6,8} {3,7,8} {1,4,7,8} {5,7,8} {2,6,7,8}Dual (rank 4)
Revlex encoding in this labeling (not canonicalized, so not linked): ***0*******0***********0****0*****************0**0*****0****0********0
Bases: {4,6,7,8} {3,6,7,8} {2,6,7,8} {1,6,7,8} {4,5,7,8} {3,5,7,8} {2,5,7,8} {1,5,7,8} {2,4,7,8} {1,4,7,8} {2,3,7,8} {1,3,7,8} {4,5,6,8} {3,5,6,8} {2,5,6,8} {1,5,6,8} {3,4,6,8} {1,4,6,8} {2,3,6,8} {1,2,6,8} {3,4,5,8} {2,4,5,8} {1,4,5,8} {2,3,5,8} {1,3,5,8} {1,2,5,8} {2,3,4,8} {1,3,4,8} {1,2,4,8} {1,2,3,8} {4,5,6,7} {3,5,6,7} {2,5,6,7} {1,5,6,7} {3,4,6,7} {2,4,6,7} {2,3,6,7} {1,3,6,7} {1,2,6,7} {3,4,5,7} {1,4,5,7} {2,3,5,7} {1,3,5,7} {1,2,5,7} {2,3,4,7} {1,3,4,7} {1,2,4,7} {1,2,3,7} {3,4,5,6} {2,4,5,6} {1,4,5,6} {1,3,5,6} {1,2,5,6} {2,3,4,6} {1,3,4,6} {1,2,4,6} {1,2,3,6} {2,3,4,5} {1,2,4,5} {1,2,3,5} {1,2,3,4}
Loops: none. Parallel classes of size > 1: none.
Downloads
- Record as JSON
- OSCAR:
matroid_from_revlex_basis_encoding("0********0****0*****0**0*****************0****0***********0*******0***", 4, 8)