Matroid 4.8.193

Label	4.8.193
Id	`r4_n8_0******00*00*********************00******************0`
Rank	4
n	8

Basic invariants

Bases	62
Circuits	32
Flats	70
Cyclic flats	10
Loops	0
Connected components	1
Automorphisms	8
Beta invariant	12
Girth	4
Simple	yes
Connected	yes
Uniform	not computed
Loopless	yes
Coloopless	yes
Paving	yes
Laminar	no
Nested	no
Self dual	yes
Identically self dual	no
Series parallel	no
Transversal	no
Supersolvable	no
Divisionally free	no
Orientable	no
Three lines	not computed

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	not computed
Components of realization space	0
Free realization space	no
Principal ideal	yes
Birational type	empty how determined: birational type method: `empty_char0`
Birational type components	show []
Good basis	not computed
Good basis v2	1,3,5,6

Geometry

Realization space: scheme simple core id	not computed
Smooth char 0	yes
Singular primes	[]
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	0

Tutte polynomial

\(T = x_{0}^{4} + 4 x_{0}^{3} + 10 x_{0}^{2} + 8 x_{0} x_{1} + 12 x_{0} + x_{1}^{4} + 4 x_{1}^{3} + 10 x_{1}^{2} + 12 x_{1}\)

Realization space

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

Combinatorics

Computed on the fly from the id.

Bases (62)

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

Non-bases (8)

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

Circuits (32)

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

Flats by rank (70)

rank 0 (1): {}
rank 1 (8): {1} {2} {3} {4} {5} {6} {7} {8}
rank 2 (28): {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}
rank 3 (32): {1,2,3,4} {2,3,5} {1,4,5} {1,3,6} {2,4,6} {1,2,5,6} {3,4,5,6} {1,2,7} {2,3,7} {1,4,7} {3,4,7} {1,3,5,7} {2,4,5,7} {1,6,7} {2,6,7} {3,6,7} {4,6,7} {1,2,8} {1,3,8} {2,4,8} {3,4,8} {1,5,8} {2,5,8} {3,5,8} {4,5,8} {2,3,6,8} {1,4,6,8} {1,7,8} {2,7,8} {3,7,8} {4,7,8} {5,6,7,8}
rank 4 (1): {1,2,3,4,5,6,7,8}

Hyperplanes (32)

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

Dual (rank 4)

Revlex encoding in this labeling (not canonicalized, so not linked): 0********************00***********************0**0*****0****0********0

Bases: {4,6,7,8} {3,6,7,8} {2,6,7,8} {1,6,7,8} {4,5,7,8} {3,5,7,8} {2,5,7,8} {1,5,7,8} {2,4,7,8} {1,4,7,8} {2,3,7,8} {1,3,7,8} {4,5,6,8} {3,5,6,8} {2,5,6,8} {1,5,6,8} {3,4,6,8} {1,4,6,8} {2,3,6,8} {1,2,6,8} {3,4,5,8} {2,4,5,8} {1,4,5,8} {2,3,5,8} {1,3,5,8} {1,2,5,8} {2,3,4,8} {1,3,4,8} {1,2,4,8} {1,2,3,8} {4,5,6,7} {3,5,6,7} {2,5,6,7} {1,5,6,7} {3,4,6,7} {2,4,6,7} {1,4,6,7} {2,3,6,7} {1,3,6,7} {1,2,6,7} {3,4,5,7} {2,4,5,7} {1,3,5,7} {1,2,5,7} {2,3,4,7} {1,3,4,7} {1,2,4,7} {1,2,3,7} {3,4,5,6} {2,4,5,6} {1,4,5,6} {2,3,5,6} {1,3,5,6} {1,2,5,6} {2,3,4,6} {1,3,4,6} {1,2,4,6} {1,2,3,6} {2,3,4,5} {1,3,4,5} {1,2,4,5} {1,2,3,5}

Loops: none. Parallel classes of size > 1: none.

Downloads

Record as JSON
OSCAR: matroid_from_revlex_basis_encoding("0********0****0*****0**0***********************00********************0", 4, 8)