Matroid 3.9.42
| Label | 3.9.42 |
|---|---|
| Id | r3_n9_0******0******0*************0******************0*****0****************0*0****0****** |
| Rank | 3 |
| n | 9 |
Basic invariants
Representability
| Characteristic set | not realizable over any field [] |
|---|---|
| Realizable | no |
| Realizable char0 | no |
| Regular | no |
| Binary | no |
| Ternary | no |
| 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": 2,
"free_rank": 2,
"torus_rank": 2,
"core_dim": 0,
"qbar_components": 1,
"type": "rational",
"reason": "free realization space"
}
] |
| Good basis | 1,2,4 |
| Good basis v2 | 1,6,7 |
Geometry
| Realization space: scheme simple core id | r3_n9_0******0******0*************0******************0*****0****************0*0****0****** |
|---|---|
| Smooth char 0 | yes |
| Singular primes | [] |
| Realization space: smooth over ℚ | not computed |
| Realization space: smooth over ℤ | yes how determined: smoothness method: three_lines_exact_bad_prime_certificate |
| Realization space: is regular scheme | yes how determined: regularity method: smooth |
| 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": 2
}
]how determined: characteristic dimensions method: cofinite_strong_groebner_candidates_exact |
| Realization space: characteristic dimension unexpected | yes |
| Realization space: characteristic dimension varies | no |
Other
| Char poly splits | false |
|---|---|
| Simplicial arrangement | no |
| Realization space: n qbar components | 1 |
Tutte polynomial
\(T = x_{0}^{3} + 6 x_{0}^{2} + 9 x_{0} x_{1} + 12 x_{0} + x_{1}^{6} + 3 x_{1}^{5} + 6 x_{1}^{4} + 10 x_{1}^{3} + 15 x_{1}^{2} + 12 x_{1}\)
Realization space
| Ring | \(\mathbb{Z}[x_{1}, x_{2}]\) |
|---|---|
| Defining ideal | \(\left(0\right)\) |
| Inequations | \(x_{1} - x_{2} \neq 0\), \(x_{2} - 1 \neq 0\), \(x_{1} \neq 0\), \(x_{1} - 1 \neq 0\), \(x_{2} \neq 0\), \(x_{1}^{2} + x_{1} x_{2}^{2} - 3 x_{1} x_{2} + x_{2} \neq 0\), \(x_{1}^{2} x_{2} - 3 x_{1} x_{2} + x_{1} + x_{2}^{2} \neq 0\) |
| Realization matrix | \(\begin{pmatrix}1 & 1 & 1 & x_{2} & 1 & 0 & 0 & x_{2} & 1 \\ 0 & 1 & x_{1} & x_{1} & x_{1} & 1 & 0 & x_{1} & 1 \\ 0 & 1 & x_{1} & x_{2} & x_{2} & 0 & 1 & x_{1} x_{2} & x_{2}\end{pmatrix}\) |
Combinatorics
Computed on the fly from the id.
Bases (75)
{1,2,4} {1,3,4} {2,3,4} {1,2,5} {1,3,5} {2,3,5} {2,4,5} {3,4,5} {1,2,6} {1,3,6} {2,3,6} {1,4,6} {3,4,6} {1,5,6} {2,5,6} {3,5,6} {4,5,6} {1,2,7} {1,3,7} {2,3,7} {1,4,7} {2,4,7} {3,4,7} {1,5,7} {2,5,7} {4,5,7} {1,6,7} {2,6,7} {3,6,7} {4,6,7} {5,6,7} {1,2,8} {1,3,8} {2,3,8} {1,4,8} {2,4,8} {3,4,8} {1,5,8} {2,5,8} {3,5,8} {4,5,8} {1,6,8} {2,6,8} {4,6,8} {5,6,8} {1,7,8} {2,7,8} {3,7,8} {5,7,8} {6,7,8} {1,2,9} {1,3,9} {2,3,9} {1,4,9} {2,4,9} {3,4,9} {1,5,9} {2,5,9} {3,5,9} {4,5,9} {1,6,9} {2,6,9} {3,6,9} {4,6,9} {1,7,9} {3,7,9} {4,7,9} {5,7,9} {6,7,9} {2,8,9} {3,8,9} {4,8,9} {5,8,9} {6,8,9} {7,8,9}Non-bases (9)
{1,2,3} {1,4,5} {2,4,6} {3,5,7} {3,6,8} {4,7,8} {5,6,9} {2,7,9} {1,8,9}Circuits (81)
{1,2,3} {1,4,5} {2,3,4,5} {2,4,6} {1,3,4,6} {1,2,5,6} {1,3,5,6} {2,3,5,6} {3,4,5,6} {1,2,4,7} {1,3,4,7} {2,3,4,7} {1,2,5,7} {3,5,7} {2,4,5,7} {1,2,6,7} {1,3,6,7} {2,3,6,7} {1,4,6,7} {3,4,6,7} {1,5,6,7} {2,5,6,7} {4,5,6,7} {1,2,4,8} {1,3,4,8} {2,3,4,8} {1,2,5,8} {1,3,5,8} {2,3,5,8} {2,4,5,8} {3,4,5,8} {1,2,6,8} {3,6,8} {1,4,6,8} {1,5,6,8} {2,5,6,8} {4,5,6,8} {1,2,7,8} {1,3,7,8} {2,3,7,8} {4,7,8} {1,5,7,8} {2,5,7,8} {1,6,7,8} {2,6,7,8} {5,6,7,8} {1,2,4,9} {1,3,4,9} {2,3,4,9} {1,2,5,9} {1,3,5,9} {2,3,5,9} {2,4,5,9} {3,4,5,9} {1,2,6,9} {1,3,6,9} {2,3,6,9} {1,4,6,9} {3,4,6,9} {5,6,9} {2,7,9} {1,3,7,9} {1,4,7,9} {3,4,7,9} {1,5,7,9} {4,5,7,9} {1,6,7,9} {3,6,7,9} {4,6,7,9} {1,8,9} {2,3,8,9} {2,4,8,9} {3,4,8,9} {2,5,8,9} {3,5,8,9} {4,5,8,9} {2,6,8,9} {4,6,8,9} {3,7,8,9} {5,7,8,9} {6,7,8,9}Flats by rank (29)
- rank 0 (1): {}
- rank 1 (9): {1} {2} {3} {4} {5} {6} {7} {8} {9}
- rank 2 (18): {1,2,3} {3,4} {2,5} {1,4,5} {1,6} {2,4,6} {1,7} {3,5,7} {6,7} {2,8} {5,8} {3,6,8} {4,7,8} {3,9} {4,9} {5,6,9} {2,7,9} {1,8,9}
- rank 3 (1): {1,2,3,4,5,6,7,8,9}
Hyperplanes (18)
{1,2,3} {3,4} {2,5} {1,4,5} {1,6} {2,4,6} {1,7} {3,5,7} {6,7} {2,8} {5,8} {3,6,8} {4,7,8} {3,9} {4,9} {5,6,9} {2,7,9} {1,8,9}Lines (9)
{1,2,3} {1,4,5} {2,4,6} {3,5,7} {3,6,8} {4,7,8} {5,6,9} {2,7,9} {1,8,9}Dual (rank 6)
Revlex encoding in this labeling (not canonicalized, so not linked): ******0****0*0****************0*****0******************0*************0******0******0
Bases: {3,5,6,7,8,9} {2,5,6,7,8,9} {1,5,6,7,8,9} {3,4,6,7,8,9} {2,4,6,7,8,9} {1,4,6,7,8,9} {1,3,6,7,8,9} {1,2,6,7,8,9} {3,4,5,7,8,9} {2,4,5,7,8,9} {1,4,5,7,8,9} {2,3,5,7,8,9} {1,2,5,7,8,9} {2,3,4,7,8,9} {1,3,4,7,8,9} {1,2,4,7,8,9} {1,2,3,7,8,9} {3,4,5,6,8,9} {2,4,5,6,8,9} {1,4,5,6,8,9} {2,3,5,6,8,9} {1,3,5,6,8,9} {1,2,5,6,8,9} {2,3,4,6,8,9} {1,3,4,6,8,9} {1,2,3,6,8,9} {2,3,4,5,8,9} {1,3,4,5,8,9} {1,2,4,5,8,9} {1,2,3,5,8,9} {1,2,3,4,8,9} {3,4,5,6,7,9} {2,4,5,6,7,9} {1,4,5,6,7,9} {2,3,5,6,7,9} {1,3,5,6,7,9} {1,2,5,6,7,9} {2,3,4,6,7,9} {1,3,4,6,7,9} {1,2,4,6,7,9} {1,2,3,6,7,9} {2,3,4,5,7,9} {1,3,4,5,7,9} {1,2,3,5,7,9} {1,2,3,4,7,9} {2,3,4,5,6,9} {1,3,4,5,6,9} {1,2,4,5,6,9} {1,2,3,4,6,9} {1,2,3,4,5,9} {3,4,5,6,7,8} {2,4,5,6,7,8} {1,4,5,6,7,8} {2,3,5,6,7,8} {1,3,5,6,7,8} {1,2,5,6,7,8} {2,3,4,6,7,8} {1,3,4,6,7,8} {1,2,4,6,7,8} {1,2,3,6,7,8} {2,3,4,5,7,8} {1,3,4,5,7,8} {1,2,4,5,7,8} {1,2,3,5,7,8} {2,3,4,5,6,8} {1,2,4,5,6,8} {1,2,3,5,6,8} {1,2,3,4,6,8} {1,2,3,4,5,8} {1,3,4,5,6,7} {1,2,4,5,6,7} {1,2,3,5,6,7} {1,2,3,4,6,7} {1,2,3,4,5,7} {1,2,3,4,5,6}
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******", 3, 9)