Matroid 4.9.67974
| Label | 4.9.67974 |
|---|---|
| Id | r4_n9_0********0****0*****0********0***********0****************0*********************************0***********0******0*****0******** |
| Rank | 4 |
| n | 9 |
Basic invariants
Representability
| Characteristic set | characteristic 0 and all primes except 2 [0,2] |
|---|---|
| Realizable | yes |
| Realizable char0 | yes |
| Regular | no |
| Binary | no |
| Ternary | no |
| Quaternary | no |
| Graphic | no |
Realization space
| Status | computed_realization_space |
|---|---|
| Dimension of realization space | 1 |
| Expected dimension over ℤ | not computed |
| Components of realization space | not computed |
| Free realization space | no |
| Principal ideal | yes |
| Birational type | genus 1 curve (53a1) how determined: birational type method: curve_genus+weierstrass_riemann_roch |
| Birational type components | show[
{
"dim": 1,
"free_rank": 0,
"torus_rank": 0,
"core_dim": 1,
"qbar_components": 1,
"type": "genus 1 curve",
"genus": 1,
"cremona_label": "53a1",
"conductor": 53,
"j_invariant": "3375/53",
"weierstrass": "[1, -1, 1, 0, 0]",
"rank": 1,
"torsion_order": 1,
"core_vars": [
"x1",
"x2"
],
"core_gens": [
"x1^2*x2 + 2*x1^2 - 3*x1*x2 + x2^2 - x1 + x2"
]
}
] |
| Good basis | not computed |
| Good basis v2 | 2,3,5,7 |
Geometry
Other
| Char poly splits | false |
|---|---|
| Simplicial arrangement | no |
| Realization space: n qbar components | not computed |
Tutte polynomial
\(T = x_{0}^{4} + 5 x_{0}^{3} + 15 x_{0}^{2} + 11 x_{0} x_{1} + 24 x_{0} + x_{1}^{5} + 4 x_{1}^{4} + 10 x_{1}^{3} + 20 x_{1}^{2} + 24 x_{1}\)
Realization space
| Ring | \(\mathbb{Z}[x_{1}, x_{2}]\) |
|---|---|
| Defining ideal | \(\left(x_{1}^{2} x_{2} + 2 x_{1}^{2} - 3 x_{1} x_{2} - x_{1} + x_{2}^{2} + x_{2}\right)\) |
| Inequations | \(2 x_{1} - 1 \neq 0\), \(2 x_{1} - x_{2} - 1 \neq 0\), \(x_{1} \neq 0\), \(x_{1} - x_{2} - 1 \neq 0\), \(x_{2} \neq 0\), \(x_{1} - x_{2} \neq 0\), \(x_{1} x_{2} + 2 x_{1} - x_{2} - 1 \neq 0\), \(x_{1} - 1 \neq 0\), \(2 x_{1}^{2} x_{2} + 2 x_{1}^{2} - x_{1} x_{2}^{2} - 3 x_{1} x_{2} - x_{1} + x_{2}^{2} + x_{2} \neq 0\), \(2 x_{1}^{3} - x_{1}^{2} - x_{1} x_{2}^{2} - 2 x_{1} x_{2} + x_{2}^{2} + x_{2} \neq 0\), \(2 x_{1}^{2} - 2 x_{1} x_{2} - x_{1} + x_{2}^{2} + x_{2} \neq 0\), \(2 x_{1}^{2} - x_{1} x_{2} - x_{1} + x_{2} \neq 0\), \(x_{1}^{2} - x_{1} + x_{2} \neq 0\), \(x_{1}^{2} - x_{2} \neq 0\), \(x_{1}^{2} - 2 x_{1} + x_{2} + 1 \neq 0\), \(x_{1}^{2} - 3 x_{1} + x_{2} + 1 \neq 0\) |
| Realization matrix | \(\begin{pmatrix}0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 1 \\ 1 & 0 & 1 & 1 & 0 & x_{1} & 0 & x_{1} - x_{2} & x_{1} \\ 1 & 0 & 0 & x_{1} & 1 & x_{1}^{2} & 0 & 2 x_{1} - x_{2} - 1 & 0 \\ 1 & 0 & 0 & x_{1} & 0 & x_{1} & 1 & 0 & x_{2}\end{pmatrix}\) |
Combinatorics
Computed on the fly from the id.
Bases (115)
{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} {2,4,5,7} {3,4,5,7} {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} {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} {1,3,6,8} {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} {2,6,7,8} {3,6,7,8} {4,6,7,8} {5,6,7,8} {1,2,3,9} {1,2,4,9} {1,3,4,9} {2,3,4,9} {1,2,5,9} {1,3,5,9} {2,3,5,9} {1,4,5,9} {2,4,5,9} {3,4,5,9} {1,2,6,9} {1,3,6,9} {2,3,6,9} {1,4,6,9} {2,4,6,9} {3,4,6,9} {1,5,6,9} {2,5,6,9} {3,5,6,9} {4,5,6,9} {1,2,7,9} {1,3,7,9} {1,4,7,9} {2,4,7,9} {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} {1,2,8,9} {1,3,8,9} {2,3,8,9} {1,4,8,9} {2,4,8,9} {3,4,8,9} {2,5,8,9} {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 (11)
{1,2,3,4} {1,2,5,6} {3,4,5,6} {1,3,5,7} {2,4,6,7} {2,3,5,8} {1,4,7,8} {2,3,7,9} {5,6,7,9} {1,5,8,9} {3,6,8,9}Circuits (82)
{1,2,3,4} {1,2,5,6} {3,4,5,6} {1,3,5,7} {1,2,4,5,7} {2,3,4,5,7} {1,2,3,6,7} {2,4,6,7} {1,3,4,6,7} {2,3,5,6,7} {1,4,5,6,7} {2,3,5,8} {1,2,4,5,8} {1,3,4,5,8} {1,2,3,6,8} {1,2,4,6,8} {1,3,4,6,8} {2,3,4,6,8} {1,3,5,6,8} {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} {2,4,5,7,8} {3,4,5,7,8} {1,2,6,7,8} {1,3,6,7,8} {2,3,6,7,8} {3,4,6,7,8} {1,5,6,7,8} {2,5,6,7,8} {3,5,6,7,8} {4,5,6,7,8} {1,2,3,5,9} {1,2,4,5,9} {1,3,4,5,9} {2,3,4,5,9} {1,2,3,6,9} {1,2,4,6,9} {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} {2,3,7,9} {1,2,4,7,9} {1,3,4,7,9} {1,2,5,7,9} {1,4,5,7,9} {2,4,5,7,9} {3,4,5,7,9} {1,2,6,7,9} {1,3,6,7,9} {1,4,6,7,9} {3,4,6,7,9} {5,6,7,9} {1,2,3,8,9} {1,2,4,8,9} {1,3,4,8,9} {2,3,4,8,9} {1,5,8,9} {2,4,5,8,9} {3,4,5,8,9} {1,2,6,8,9} {3,6,8,9} {1,4,6,8,9} {2,4,6,8,9} {2,5,6,8,9} {4,5,6,8,9} {1,2,7,8,9} {1,3,7,8,9} {2,4,7,8,9} {3,4,7,8,9} {2,5,7,8,9} {3,5,7,8,9} {4,5,7,8,9} {1,6,7,8,9} {2,6,7,8,9} {4,6,7,8,9}Flats by rank (98)
- rank 0 (1): {}
- rank 1 (9): {1} {2} {3} {4} {5} {6} {7} {8} {9}
- rank 2 (36): {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} {1,9} {2,9} {3,9} {4,9} {5,9} {6,9} {7,9} {8,9}
- rank 3 (51): {1,2,3,4} {1,4,5} {2,4,5} {1,3,6} {2,3,6} {1,4,6} {1,2,5,6} {3,4,5,6} {1,2,7} {3,4,7} {2,5,7} {1,3,5,7} {4,5,7} {1,6,7} {3,6,7} {2,4,6,7} {1,2,8} {1,3,8} {2,4,8} {3,4,8} {2,3,5,8} {4,5,8} {1,6,8} {2,6,8} {4,6,8} {5,6,8} {2,7,8} {3,7,8} {1,4,7,8} {5,7,8} {6,7,8} {1,2,9} {1,3,9} {1,4,9} {2,4,9} {3,4,9} {2,5,9} {3,5,9} {4,5,9} {1,6,9} {2,6,9} {4,6,9} {1,7,9} {2,3,7,9} {4,7,9} {5,6,7,9} {2,8,9} {4,8,9} {1,5,8,9} {3,6,8,9} {7,8,9}
- rank 4 (1): {1,2,3,4,5,6,7,8,9}
Hyperplanes (51)
{1,2,3,4} {1,4,5} {2,4,5} {1,3,6} {2,3,6} {1,4,6} {1,2,5,6} {3,4,5,6} {1,2,7} {3,4,7} {2,5,7} {1,3,5,7} {4,5,7} {1,6,7} {3,6,7} {2,4,6,7} {1,2,8} {1,3,8} {2,4,8} {3,4,8} {2,3,5,8} {4,5,8} {1,6,8} {2,6,8} {4,6,8} {5,6,8} {2,7,8} {3,7,8} {1,4,7,8} {5,7,8} {6,7,8} {1,2,9} {1,3,9} {1,4,9} {2,4,9} {3,4,9} {2,5,9} {3,5,9} {4,5,9} {1,6,9} {2,6,9} {4,6,9} {1,7,9} {2,3,7,9} {4,7,9} {5,6,7,9} {2,8,9} {4,8,9} {1,5,8,9} {3,6,8,9} {7,8,9}Dual (rank 5)
Revlex encoding in this labeling (not canonicalized, so not linked): ********0*****0******0***********0*********************************0****************0***********0********0*****0****0********0
Bases: {4,6,7,8,9} {3,6,7,8,9} {2,6,7,8,9} {1,6,7,8,9} {4,5,7,8,9} {3,5,7,8,9} {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} {4,5,6,8,9} {3,5,6,8,9} {2,5,6,8,9} {1,5,6,8,9} {3,4,6,8,9} {1,4,6,8,9} {2,3,6,8,9} {1,3,6,8,9} {1,2,6,8,9} {3,4,5,8,9} {2,4,5,8,9} {1,4,5,8,9} {2,3,5,8,9} {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} {4,5,6,7,9} {3,5,6,7,9} {2,5,6,7,9} {1,5,6,7,9} {3,4,6,7,9} {2,4,6,7,9} {2,3,6,7,9} {1,3,6,7,9} {1,2,6,7,9} {3,4,5,7,9} {2,4,5,7,9} {1,4,5,7,9} {2,3,5,7,9} {1,3,5,7,9} {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} {3,4,5,6,9} {2,4,5,6,9} {1,4,5,6,9} {1,3,5,6,9} {1,2,5,6,9} {2,3,4,6,9} {1,3,4,6,9} {1,2,4,6,9} {1,2,3,6,9} {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} {4,5,6,7,8} {3,5,6,7,8} {2,5,6,7,8} {1,5,6,7,8} {3,4,6,7,8} {2,4,6,7,8} {1,4,6,7,8} {2,3,6,7,8} {1,3,6,7,8} {1,2,6,7,8} {3,4,5,7,8} {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} {3,4,5,6,8} {2,4,5,6,8} {2,3,5,6,8} {1,3,5,6,8} {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} {3,4,5,6,7} {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} {1,3,4,6,7} {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: none.
Downloads
- Record as JSON
- OSCAR:
matroid_from_revlex_basis_encoding("0********0****0*****0********0***********0****************0*********************************0***********0******0*****0********", 4, 9)