Matroid 3.5.6
| Label | 3.5.6 |
|---|---|
| Id | r3_n5_0000****** |
| Rank | 3 |
| n | 5 |
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 | 1 |
| Expected dimension over ℤ | 2 |
| Components of realization space | 1 |
| Free realization space | yes |
| Principal ideal | yes |
| Birational type | rational how determined: birational type method: free_presentation |
| Birational type components | show[
{
"dim": 1,
"free_rank": 1,
"torus_rank": 1,
"core_dim": 0,
"qbar_components": 1,
"type": "rational",
"reason": "free realization space"
}
] |
| Good basis | not computed |
| Good basis v2 | 1,2,5 |
Geometry
| Realization space: scheme simple core id | r3_n5_0000****** |
|---|---|
| Smooth char 0 | not computed |
| Singular primes | not computed |
| Realization space: smooth over ℚ | not computed |
| Realization space: smooth over ℤ | yes how determined: smoothness method: simple_disconnected_open_affine |
| Realization space: is regular scheme | yes how determined: regularity method: simple_disconnected_open_affine |
| Realization space: singular fiber primes | [] how determined: singular fiber primes method: smooth_over_ZZ |
| Realization space: singular fiber characteristics | not computed |
| Realization space: characteristic dimensions | show[
{
"p": 0,
"d": 1
}
]how determined: characteristic dimensions method: cofinite_smooth_equidimensional |
| Realization space: characteristic dimension unexpected | no |
| Realization space: characteristic dimension varies | no |
Other
| Char poly splits | true |
|---|---|
| Simplicial arrangement | yes |
| Realization space: n qbar components | 1 |
Tutte polynomial
\(T = x^{3} + 2 x^{2} + x y^{2} + 2 x y\)
Realization space
| Ring | \(\mathbb{Z}[x_{1}]\) |
|---|---|
| Defining ideal | \(\left(0\right)\) |
| Inequations | \(x_{1} \neq 0\), \(x_{1} - 1 \neq 0\) |
| Realization matrix | \(\begin{pmatrix}1 & 0 & 1 & 1 & 0 \\ 0 & 1 & 1 & x_{1} & 0 \\ 0 & 0 & 0 & 0 & 1\end{pmatrix}\) |
Combinatorics
Computed on the fly from the id.
Bases (6)
{1,2,5} {1,3,5} {2,3,5} {1,4,5} {2,4,5} {3,4,5}Non-bases (4)
{1,2,3} {1,2,4} {1,3,4} {2,3,4}Circuits (4)
{1,2,3} {1,2,4} {1,3,4} {2,3,4}Flats by rank (12)
- rank 0 (1): {}
- rank 1 (5): {1} {2} {3} {4} {5}
- rank 2 (5): {1,2,3,4} {1,5} {2,5} {3,5} {4,5}
- rank 3 (1): {1,2,3,4,5}
Hyperplanes (5)
{1,2,3,4} {1,5} {2,5} {3,5} {4,5}Lines (1)
{1,2,3,4}Dual (rank 2)
Revlex encoding in this labeling (not canonicalized, so not linked): ******0000
Bases: {3,4} {2,4} {1,4} {2,3} {1,3} {1,2}
Loops: none. Parallel classes of size > 1: none.
Downloads
- Record as JSON
- OSCAR:
matroid_from_revlex_basis_encoding("0000******", 3, 5)