Matroid database

Matroid 4.6.4

Label4.6.4
Idr4_n6_00***0*********
Rank4
n6

Basic invariants

Bases12
Circuits4
Flats34
Cyclic flatsnot computed
Loops0
Connected components1
Automorphismsnot computed
Beta invariantnot computed
Girth3
Simpleyes
Connectedyes
Uniformno
Looplessyes
Colooplessyes
Pavingnot computed
Laminarnot computed
Nestednot computed
Self dualnot computed
Identically self dualnot computed
Series parallelnot computed
Transversalnot computed
Supersolvableno
Divisionally freeno
Orientablenot computed
Three linesnot computed

Representability

Characteristic setcharacteristic 0 and all primes [0]
Realizableyes
Realizable char0yes
Regularno
Binaryno
Ternaryyes
Quaternaryyes
Graphicno

Realization space

Statuscomputed_realization_space
Dimension of realization space1
Expected dimension over ℤnot computed
Components of realization space1
Free realization spaceyes
Principal idealyes
Birational typerational
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 basisnot computed
Good basis v21,2,4,5

Geometry

Realization space: scheme simple core idnot computed
Smooth char 0not computed
Singular primesnot computed
Realization space: smooth over ℚnot computed
Realization space: smooth over ℤnot computed
Realization space: is regular schemenot computed
Realization space: singular fiber primesnot computed
Realization space: singular fiber characteristicsnot computed
Realization space: characteristic dimensionsnot computed
Realization space: characteristic dimension unexpectednot computed
Realization space: characteristic dimension variesnot computed

Other

Char poly splitsfalse
Simplicial arrangementno
Realization space: n qbar components1

Tutte polynomial

\(T = x^{4} + 2 x^{3} + x^{2} y + 2 x^{2} + x y + 2 x + y^{2} + 2 y\)

Realization space

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

Combinatorics

Computed on the fly from the id.

Bases (12){1,2,4,5} {1,3,4,5} {2,3,4,5} {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}
Non-bases (3){1,2,3,4} {1,2,3,5} {1,2,3,6}
Circuits (4){1,2,3} {1,2,4,5,6} {1,3,4,5,6} {2,3,4,5,6}
Flats by rank (34)
Hyperplanes (13){1,2,3,4} {1,2,3,5} {1,4,5} {2,4,5} {3,4,5} {1,2,3,6} {1,4,6} {2,4,6} {3,4,6} {1,5,6} {2,5,6} {3,5,6} {4,5,6}
Dual (rank 2)

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

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

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

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