Matroid 4.9.184684
| Label | 4.9.184684 |
|---|---|
| Id | r4_n9_0000000**0****000**0****0000*******00**0****00**00*****0****0**00*****00**0****00*********0**00*********00*****00****0******** |
| Rank | 4 |
| n | 9 |
Basic invariants
Representability
| Characteristic set | characteristic 0 and all primes [0] |
|---|---|
| Realizable | yes |
| Realizable char0 | yes |
| Regular | no |
| Binary | no |
| Ternary | yes |
| Quaternary | yes |
| Graphic | no |
Realization space
| Status | computed_realization_space |
|---|---|
| Dimension of realization space | 3 |
| Expected dimension over ℤ | not computed |
| Components of realization space | not computed |
| 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 | 1,3,4,9 |
Geometry
Other
| Char poly splits | false |
|---|---|
| Simplicial arrangement | no |
| Realization space: n qbar components | not computed |
Tutte polynomial
\(T = x_{0}^{4} + x_{0}^{3} x_{1} + 4 x_{0}^{3} + 5 x_{0}^{2} x_{1} + 9 x_{0}^{2} + 6 x_{0} x_{1}^{2} + 15 x_{0} x_{1} + 8 x_{0} + x_{1}^{5} + 4 x_{1}^{4} + 10 x_{1}^{3} + 14 x_{1}^{2} + 8 x_{1}\)
Realization space
| Ring | \(\mathbb{Z}[x_{1}]\) |
|---|---|
| Defining ideal | \(\left(x_{1}^{2} - x_{1} + 1\right)\) |
| Inequations | none |
| Realization matrix | \(\begin{pmatrix}1 & 1 & 0 & 0 & 0 & 1 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 & 1 & 1 & 0 & x_{1} & 0 \\ 0 & 0 & 0 & 1 & 1 & x_{1} & x_{1} - 1 & x_{1} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & x_{1} & x_{1} & 1\end{pmatrix}\) |
Combinatorics
Computed on the fly from the id.
Bases (86)
{1,3,4,6} {2,3,4,6} {1,3,5,6} {2,3,5,6} {1,4,5,6} {2,4,5,6} {1,3,4,7} {2,3,4,7} {1,3,5,7} {2,3,5,7} {1,4,5,7} {2,4,5,7} {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} {1,3,4,8} {2,3,4,8} {1,3,5,8} {2,3,5,8} {1,4,5,8} {2,4,5,8} {1,3,6,8} {2,3,6,8} {3,4,6,8} {1,5,6,8} {2,5,6,8} {3,5,6,8} {4,5,6,8} {1,3,7,8} {2,3,7,8} {1,4,7,8} {2,4,7,8} {1,5,7,8} {2,5,7,8} {1,6,7,8} {2,6,7,8} {3,6,7,8} {4,6,7,8} {5,6,7,8} {1,3,4,9} {2,3,4,9} {1,3,5,9} {2,3,5,9} {1,4,5,9} {2,4,5,9} {1,3,6,9} {2,3,6,9} {1,4,6,9} {2,4,6,9} {3,4,6,9} {1,5,6,9} {2,5,6,9} {3,5,6,9} {4,5,6,9} {1,3,7,9} {2,3,7,9} {3,4,7,9} {1,5,7,9} {2,5,7,9} {3,5,7,9} {4,5,7,9} {1,6,7,9} {2,6,7,9} {3,6,7,9} {4,6,7,9} {1,3,8,9} {2,3,8,9} {1,4,8,9} {2,4,8,9} {3,4,8,9} {3,5,8,9} {4,5,8,9} {1,6,8,9} {2,6,8,9} {4,6,8,9} {5,6,8,9} {1,7,8,9} {2,7,8,9} {3,7,8,9} {4,7,8,9} {5,7,8,9} {6,7,8,9}Non-bases (40)
{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} {1,2,5,6} {3,4,5,6} {1,2,3,7} {1,2,4,7} {1,2,5,7} {3,4,5,7} {1,2,6,7} {1,3,6,7} {2,3,6,7} {1,2,3,8} {1,2,4,8} {1,2,5,8} {3,4,5,8} {1,2,6,8} {1,4,6,8} {2,4,6,8} {1,2,7,8} {3,4,7,8} {3,5,7,8} {4,5,7,8} {1,2,3,9} {1,2,4,9} {1,2,5,9} {3,4,5,9} {1,2,6,9} {1,2,7,9} {1,4,7,9} {2,4,7,9} {5,6,7,9} {1,2,8,9} {1,5,8,9} {2,5,8,9} {3,6,8,9}Circuits (38)
{1,2} {3,4,5} {1,3,6,7} {2,3,6,7} {1,4,5,6,7} {2,4,5,6,7} {1,4,6,8} {2,4,6,8} {1,3,5,6,8} {2,3,5,6,8} {3,4,7,8} {3,5,7,8} {4,5,7,8} {1,5,6,7,8} {2,5,6,7,8} {1,3,4,6,9} {2,3,4,6,9} {1,3,5,6,9} {2,3,5,6,9} {1,4,5,6,9} {2,4,5,6,9} {1,4,7,9} {2,4,7,9} {1,3,5,7,9} {2,3,5,7,9} {3,4,6,7,9} {5,6,7,9} {1,3,4,8,9} {2,3,4,8,9} {1,5,8,9} {2,5,8,9} {3,6,8,9} {4,5,6,8,9} {1,3,7,8,9} {2,3,7,8,9} {1,6,7,8,9} {2,6,7,8,9} {4,6,7,8,9}Flats by rank (59)
- rank 0 (1): {}
- rank 1 (8): {1,2} {3} {4} {5} {6} {7} {8} {9}
- rank 2 (26): {1,2,3} {1,2,4} {1,2,5} {3,4,5} {1,2,6} {3,6} {4,6} {5,6} {1,2,7} {3,7} {4,7} {5,7} {6,7} {1,2,8} {3,8} {4,8} {5,8} {6,8} {7,8} {1,2,9} {3,9} {4,9} {5,9} {6,9} {7,9} {8,9}
- rank 3 (23): {1,2,3,4,5} {1,2,5,6} {3,4,5,6} {1,2,5,7} {1,2,3,6,7} {4,6,7} {1,2,3,8} {1,2,4,6,8} {5,6,8} {1,2,7,8} {3,4,5,7,8} {6,7,8} {1,2,3,9} {3,4,5,9} {1,2,6,9} {4,6,9} {3,7,9} {1,2,4,7,9} {5,6,7,9} {4,8,9} {1,2,5,8,9} {3,6,8,9} {7,8,9}
- rank 4 (1): {1,2,3,4,5,6,7,8,9}
Hyperplanes (23)
{1,2,3,4,5} {1,2,5,6} {3,4,5,6} {1,2,5,7} {1,2,3,6,7} {4,6,7} {1,2,3,8} {1,2,4,6,8} {5,6,8} {1,2,7,8} {3,4,5,7,8} {6,7,8} {1,2,3,9} {3,4,5,9} {1,2,6,9} {4,6,9} {3,7,9} {1,2,4,7,9} {5,6,7,9} {4,8,9} {1,2,5,8,9} {3,6,8,9} {7,8,9}Dual (rank 5)
Revlex encoding in this labeling (not canonicalized, so not linked): ********0****00*****00*********00**0*********00****0**00*****00**0****0*****00**00****0**00*******0000****0**000****0**0000000
Bases: {2,5,7,8,9} {1,5,7,8,9} {2,4,7,8,9} {1,4,7,8,9} {2,3,7,8,9} {1,3,7,8,9} {2,5,6,8,9} {1,5,6,8,9} {2,4,6,8,9} {1,4,6,8,9} {2,3,6,8,9} {1,3,6,8,9} {2,3,5,8,9} {1,3,5,8,9} {1,2,5,8,9} {2,3,4,8,9} {1,3,4,8,9} {1,2,4,8,9} {1,2,3,8,9} {2,5,6,7,9} {1,5,6,7,9} {2,4,6,7,9} {1,4,6,7,9} {2,3,6,7,9} {1,3,6,7,9} {2,4,5,7,9} {1,4,5,7,9} {1,2,5,7,9} {2,3,4,7,9} {1,3,4,7,9} {1,2,4,7,9} {1,2,3,7,9} {2,4,5,6,9} {1,4,5,6,9} {2,3,5,6,9} {1,3,5,6,9} {2,3,4,6,9} {1,3,4,6,9} {2,3,4,5,9} {1,3,4,5,9} {1,2,4,5,9} {1,2,3,5,9} {1,2,3,4,9} {2,5,6,7,8} {1,5,6,7,8} {2,4,6,7,8} {1,4,6,7,8} {2,3,6,7,8} {1,3,6,7,8} {2,4,5,7,8} {1,4,5,7,8} {2,3,5,7,8} {1,3,5,7,8} {1,2,5,7,8} {2,3,4,7,8} {1,3,4,7,8} {1,2,4,7,8} {1,2,3,7,8} {2,4,5,6,8} {1,4,5,6,8} {1,2,5,6,8} {2,3,4,6,8} {1,3,4,6,8} {1,2,4,6,8} {1,2,3,6,8} {2,3,4,5,8} {1,3,4,5,8} {1,2,4,5,8} {1,2,3,5,8} {2,4,5,6,7} {1,4,5,6,7} {2,3,5,6,7} {1,3,5,6,7} {1,2,5,6,7} {1,2,4,6,7} {1,2,3,6,7} {2,3,4,5,7} {1,3,4,5,7} {1,2,3,5,7} {1,2,3,4,7} {2,3,4,5,6} {1,3,4,5,6} {1,2,4,5,6} {1,2,3,5,6} {1,2,3,4,6} {1,2,3,4,5}
Loops: none. Parallel classes of size > 1: {1,2}.
Downloads
- Record as JSON
- OSCAR:
matroid_from_revlex_basis_encoding("0000000**0****000**0****0000*******00**0****00**00*****0****0**00*****00**0****00*********0**00*********00*****00****0********", 4, 9)