Matroid 4.9.174230

Label	4.9.174230
Id	`r4_n9_000000*******0*****000***0********00****0*****0*********0**********0*********0***0**`
Rank	4
n	9

Basic invariants

Bases	107
Circuits	66
Flats	87
Cyclic flats	12
Loops	0
Connected components	1
Automorphisms	1
Beta invariant	19
Girth	3
Simple	yes
Connected	yes
Uniform	not computed
Loopless	yes
Coloopless	yes
Paving	no
Laminar	no
Nested	no
Self dual	no
Identically self dual	no
Series parallel	no
Transversal	no
Supersolvable	no
Divisionally free	no
Orientable	no
Three lines	not computed

Representability

Characteristic set	characteristic 0 and all primes `[0]`
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	3
Expected dimension over ℤ	not computed
Components of realization space	not computed
Free realization space	no
Principal ideal	yes
Birational type	rational over Q(a)/(a^2 - a + 1) how determined: birational type method: `point_field`
Birational type components	show [ { "dim": 1, "free_rank": 1, "torus_rank": 1, "core_dim": 0, "qbar_components": 2, "type": "rational over Q(a)/(a^2 - a + 1)", "field": "Q(a)/(a^2 - a + 1)", "minpoly": "T^2 - T + 1", "degree": 2, "reason": "A^1 over the field of definition", "core_vars": [ "x1" ], "core_gens": [ "x1^2 - x1 + 1" ] } ]
Good basis	not computed
Good basis v2	2,3,4,8

Geometry

Realization space: scheme simple core id	not computed
Smooth char 0	yes
Singular primes	[3]
Realization space: smooth over ℚ	not computed
Realization space: smooth over ℤ	not computed
Realization space: is regular scheme	not computed
Realization space: singular fiber primes	not computed
Realization space: singular fiber characteristics	not computed
Realization space: characteristic dimensions	not computed
Realization space: characteristic dimension unexpected	not computed
Realization space: characteristic dimension varies	not computed

Other

Char poly splits	false
Simplicial arrangement	no
Realization space: n qbar components	not computed

Tutte polynomial

\(T = x_{0}^{4} + 5 x_{0}^{3} + x_{0}^{2} x_{1} + 14 x_{0}^{2} + 2 x_{0} x_{1}^{2} + 13 x_{0} x_{1} + 19 x_{0} + x_{1}^{5} + 4 x_{1}^{4} + 10 x_{1}^{3} + 18 x_{1}^{2} + 19 x_{1}\)

Realization space

Ring	\(\mathbb{Z}[x_{1}, x_{2}]\)
Defining ideal	\(\left(x_{1}^{2} - x_{1} + 1\right)\)
Inequations	\(x_{2} \neq 0\), \(x_{2} - 1 \neq 0\), \(x_{1} + x_{2} - 1 \neq 0\), \(x_{1}^{2} x_{2} - x_{1}^{2} - 2 x_{1} x_{2} + x_{1} + x_{2} - 1 \neq 0\), \(x_{1}^{2} + x_{1} x_{2} - x_{1} - x_{2} + 1 \neq 0\), \(x_{1}^{3} + x_{1}^{2} x_{2} - 2 x_{1}^{2} - 2 x_{1} x_{2} + 3 x_{1} + x_{2} - 1 \neq 0\), \(x_{1}^{2} x_{2} - 2 x_{1} x_{2} + x_{2} + 1 \neq 0\), \(x_{1} x_{2} - x_{1} - x_{2} \neq 0\), \(x_{1} x_{2} + x_{2} - 1 \neq 0\), \(x_{1} x_{2} - 1 \neq 0\), \(x_{1} x_{2} - x_{2} - 1 \neq 0\), \(x_{1}^{3} x_{2} - x_{1}^{2} x_{2} - x_{1}^{2} + x_{1} x_{2} + x_{1} - x_{2} - 1 \neq 0\), \(x_{1}^{2} + x_{1} x_{2} - x_{1} + 1 \neq 0\), \(x_{1}^{2} x_{2} - x_{1} + 1 \neq 0\)
Realization matrix	\(\begin{pmatrix}1 & 1 & 0 & 0 & 1 & x_{2} & 0 & 0 & 1 \\ 1 & 0 & 1 & 0 & x_{1} x_{2} & x_{2} & x_{2} & 0 & x_{2} \\ 0 & 0 & 0 & 1 & 1 & -x_{1} + 1 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & x_{2} & x_{1} x_{2} & 1 & x_{2}\end{pmatrix}\)

Combinatorics

Computed on the fly from the id.

Bases (107)

{1,2,4,6} {1,3,4,6} {2,3,4,6} {1,2,5,6} {1,3,5,6} {2,3,5,6} {1,4,5,6} {2,4,5,6} {3,4,5,6} {1,2,4,7} {1,3,4,7} {2,3,4,7} {1,2,5,7} {1,3,5,7} {2,3,5,7} {1,4,5,7} {2,4,5,7} {3,4,5,7} {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} {1,2,4,8} {1,3,4,8} {2,3,4,8} {1,2,5,8} {1,3,5,8} {2,3,5,8} {1,4,5,8} {2,4,5,8} {3,4,5,8} {1,2,6,8} {1,3,6,8} {2,3,6,8} {2,4,6,8} {3,4,6,8} {1,5,6,8} {3,5,6,8} {4,5,6,8} {1,2,7,8} {1,3,7,8} {2,3,7,8} {1,4,7,8} {2,4,7,8} {1,5,7,8} {2,5,7,8} {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} {1,2,4,9} {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} {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} {2,3,7,9} {1,4,7,9} {2,4,7,9} {3,4,7,9} {1,5,7,9} {2,5,7,9} {3,5,7,9} {1,6,7,9} {2,6,7,9} {3,6,7,9} {4,6,7,9} {5,6,7,9} {1,2,8,9} {1,3,8,9} {2,3,8,9} {1,4,8,9} {2,4,8,9} {3,4,8,9} {1,5,8,9} {2,5,8,9} {4,5,8,9} {1,6,8,9} {2,6,8,9} {3,6,8,9} {4,6,8,9} {5,6,8,9} {1,7,8,9} {3,7,8,9} {4,7,8,9} {5,7,8,9} {6,7,8,9}

Non-bases (19)

{1,2,3,4} {1,2,3,5} {1,2,4,5} {1,3,4,5} {2,3,4,5} {1,2,3,6} {1,2,3,7} {1,2,6,7} {1,3,6,7} {2,3,6,7} {1,2,3,8} {1,4,6,8} {2,5,6,8} {3,4,7,8} {1,2,3,9} {2,4,6,9} {4,5,7,9} {3,5,8,9} {2,7,8,9}

Circuits (66)

{1,2,3} {1,2,4,5} {1,3,4,5} {2,3,4,5} {1,2,6,7} {1,3,6,7} {2,3,6,7} {1,4,5,6,7} {2,4,5,6,7} {3,4,5,6,7} {1,4,6,8} {2,3,4,6,8} {2,5,6,8} {1,3,5,6,8} {3,4,5,6,8} {1,2,4,7,8} {3,4,7,8} {1,2,5,7,8} {1,3,5,7,8} {2,3,5,7,8} {1,4,5,7,8} {2,4,5,7,8} {2,4,6,7,8} {1,5,6,7,8} {3,5,6,7,8} {4,5,6,7,8} {2,4,6,9} {1,3,4,6,9} {1,2,5,6,9} {1,3,5,6,9} {2,3,5,6,9} {1,4,5,6,9} {3,4,5,6,9} {1,2,4,7,9} {1,3,4,7,9} {2,3,4,7,9} {1,2,5,7,9} {1,3,5,7,9} {2,3,5,7,9} {4,5,7,9} {1,4,6,7,9} {3,4,6,7,9} {1,5,6,7,9} {2,5,6,7,9} {3,5,6,7,9} {1,2,4,8,9} {1,3,4,8,9} {2,3,4,8,9} {1,2,5,8,9} {3,5,8,9} {1,4,5,8,9} {2,4,5,8,9} {1,2,6,8,9} {1,3,6,8,9} {2,3,6,8,9} {3,4,6,8,9} {1,5,6,8,9} {4,5,6,8,9} {2,7,8,9} {1,3,7,8,9} {1,4,7,8,9} {1,5,7,8,9} {1,6,7,8,9} {3,6,7,8,9} {4,6,7,8,9} {5,6,7,8,9}

Flats by rank (87)

rank 0 (1): {}
rank 1 (9): {1} {2} {3} {4} {5} {6} {7} {8} {9}
rank 2 (34): {1,2,3} {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 (42): {1,2,3,4,5} {3,4,6} {1,5,6} {3,5,6} {4,5,6} {1,4,7} {2,4,7} {1,5,7} {2,5,7} {3,5,7} {1,2,3,6,7} {4,6,7} {5,6,7} {1,2,3,8} {2,4,8} {1,5,8} {4,5,8} {3,6,8} {1,4,6,8} {2,5,6,8} {1,7,8} {3,4,7,8} {5,7,8} {6,7,8} {1,2,3,9} {1,4,9} {3,4,9} {1,5,9} {2,5,9} {1,6,9} {3,6,9} {2,4,6,9} {5,6,9} {1,7,9} {3,7,9} {4,5,7,9} {6,7,9} {1,8,9} {4,8,9} {3,5,8,9} {6,8,9} {2,7,8,9}
rank 4 (1): {1,2,3,4,5,6,7,8,9}

Hyperplanes (42)

{1,2,3,4,5} {3,4,6} {1,5,6} {3,5,6} {4,5,6} {1,4,7} {2,4,7} {1,5,7} {2,5,7} {3,5,7} {1,2,3,6,7} {4,6,7} {5,6,7} {1,2,3,8} {2,4,8} {1,5,8} {4,5,8} {3,6,8} {1,4,6,8} {2,5,6,8} {1,7,8} {3,4,7,8} {5,7,8} {6,7,8} {1,2,3,9} {1,4,9} {3,4,9} {1,5,9} {2,5,9} {1,6,9} {3,6,9} {2,4,6,9} {5,6,9} {1,7,9} {3,7,9} {4,5,7,9} {6,7,9} {1,8,9} {4,8,9} {3,5,8,9} {6,8,9} {2,7,8,9}

Dual (rank 5)

Revlex encoding in this labeling (not canonicalized, so not linked): ****0*******0*************0**************0*************0*********0*******0***0************0*******000*********0*********000000

Bases: {3,5,7,8,9} {2,5,7,8,9} {1,5,7,8,9} {3,4,7,8,9} {2,4,7,8,9} {1,4,7,8,9} {2,3,7,8,9} {1,3,7,8,9} {1,2,7,8,9} {3,5,6,8,9} {2,5,6,8,9} {1,5,6,8,9} {3,4,6,8,9} {2,4,6,8,9} {1,4,6,8,9} {2,3,6,8,9} {1,3,6,8,9} {1,2,6,8,9} {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} {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} {1,4,6,7,9} {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} {1,3,5,7,9} {1,2,5,7,9} {2,3,4,7,9} {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} {2,3,5,6,9} {1,3,5,6,9} {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} {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} {1,2,5,7,8} {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} {1,4,5,6,8} {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} {2,3,4,5,8} {1,3,4,5,8} {1,2,4,5,8} {1,2,3,5,8} {1,2,3,4,8} {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} {2,3,4,6,7} {1,3,4,6,7} {1,2,3,6,7} {2,3,4,5,7} {1,3,4,5,7} {1,2,4,5,7} {1,2,3,5,7} {1,2,3,4,7} {2,3,4,5,6} {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("000000*********0*********000*******0************0***0*******0*********0*************0**************0*************0*******0****", 4, 9)