Matroid database

Matroid 3.5.11

Label3.5.11
Idr3_n5_00000000**
Rank3
n5
2,345

Affine diagram (real realization). Loops (not drawn): 1.

23451

A graph with this cycle matroid; edge labels are elements.

Basic invariants

Bases2
Circuits2
Flats8
Cyclic flatsnot computed
Loops1
Connected components4
Automorphismsnot computed
Beta invariantnot computed
Girth1
Simpleno
Connectedno
Uniformno
Looplessno
Colooplessno
Pavingno
Laminarnot computed
Nestednot computed
Self dualnot computed
Identically self dualnot computed
Series parallelnot computed
Transversalnot computed
Supersolvableyes
Divisionally freeyes
Orientableyes
Three linesnot computed

Representability

Characteristic setcharacteristic 0 and all primes [0]
Realizableyes
Realizable char0yes
Regularyes
Binaryyes
Ternaryyes
Quaternaryyes
Graphicyes

Realization space

Statuscomputed_realization_space
Dimension of realization space0
Expected dimension over ℤ1
Components of realization space1
Free realization spaceyes
Principal idealyes
Birational typerational
how determined: birational type method: rigid
Birational type components
show
[
 {
  "dim": 0,
  "free_rank": 0,
  "torus_rank": 0,
  "core_dim": 0,
  "qbar_components": 1,
  "type": "rational",
  "reason": "rigid: unique realization over Q (integer matrix)"
 }
]
Good basisnot computed
Good basis v22,4,5

Geometry

Realization space: scheme simple core idr3_n3_*
Smooth char 0not computed
Singular primesnot computed
Realization space: smooth over ℚnot computed
Realization space: smooth over ℤyes
how determined: smoothness method: simple_core smoothness witness: r3_n3_*
Realization space: is regular schemeyes
how determined: regularity method: simple_core regularity witness: r3_n3_*
Realization space: singular fiber primes[]
how determined: singular fiber primes method: simple_core
Realization space: singular fiber characteristicsnot computed
Realization space: characteristic dimensions
show
[
 {
  "p": 0,
  "d": 0
 }
]
how determined: characteristic dimensions method: simple_core
Realization space: characteristic dimension unexpectedno
Realization space: characteristic dimension variesno

Other

Char poly splitstrue
Simplicial arrangementyes
Realization space: n qbar components1

Tutte polynomial

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

Realization space

Ring\(\mathbb{Z}\)
Defining ideal\(\left(0\right)\)
Inequationsnone
Realization matrix\(\begin{pmatrix}0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1\end{pmatrix}\)

Combinatorics

Computed on the fly from the id.

Bases (2){2,4,5} {3,4,5}
Non-bases (8){1,2,3} {1,2,4} {1,3,4} {2,3,4} {1,2,5} {1,3,5} {2,3,5} {1,4,5}
Circuits (2){1} {2,3}
Flats by rank (8)
Hyperplanes (3){1,2,3,4} {1,2,3,5} {1,4,5}
Lines (0)
Dual (rank 2)

Revlex encoding in this labeling (not canonicalized, so not linked): **00000000

Bases: {1,3} {1,2}

Loops: 1. Parallel classes of size > 1: {2,3}.

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