Matroid database

Matroid 3.5.0

Label3.5.0
Idr3_n5_**********
Rank3
n5
12345

Affine diagram (real realization).

Basic invariants

Bases10
Circuits5
Flats17
Cyclic flatsnot computed
Loops0
Connected components1
Automorphismsnot computed
Beta invariantnot computed
Girth4
Simpleyes
Connectedyes
Uniformyes
Looplessyes
Colooplessyes
Pavingyes
Laminarnot computed
Nestednot computed
Self dualnot computed
Identically self dualnot computed
Series parallelnot computed
Transversalnot computed
Supersolvableno
Divisionally freeno
Orientableyes
Three linesnot computed

Representability

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

Realization space

Statuscomputed_realization_space
Dimension of realization space2
Expected dimension over ℤ3
Components of realization space1
Free realization spaceyes
Principal idealyes
Birational typerational
how determined: birational type method: free_presentation
Birational type components
show
[
 {
  "dim": 2,
  "free_rank": 2,
  "torus_rank": 2,
  "core_dim": 0,
  "qbar_components": 1,
  "type": "rational",
  "reason": "free realization space"
 }
]
Good basisnot computed
Good basis v21,2,3

Geometry

Realization space: scheme simple core idr3_n5_**********
Smooth char 0not computed
Singular primesnot computed
Realization space: smooth over ℚnot computed
Realization space: smooth over ℤyes
how determined: smoothness method: three_lines_deletion smoothness witness: 1=>r3_n4_****
Realization space: is regular schemeyes
how determined: regularity method: three_lines_deletion regularity witness: 1=>r3_n4_****
Realization space: singular fiber primes[]
how determined: singular fiber primes method: smooth_over_ZZ
Realization space: singular fiber characteristicsnot computed
Realization space: characteristic dimensions
show
[
 {
  "p": 0,
  "d": 2
 }
]
how determined: characteristic dimensions method: cofinite_smooth_equidimensional
Realization space: characteristic dimension unexpectedno
Realization space: characteristic dimension variesno

Other

Char poly splitsfalse
Simplicial arrangementno
Realization space: n qbar components1

Tutte polynomial

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

Realization space

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

Combinatorics

Computed on the fly from the id.

Bases (10){1,2,3} {1,2,4} {1,3,4} {2,3,4} {1,2,5} {1,3,5} {2,3,5} {1,4,5} {2,4,5} {3,4,5}
Non-bases (0)
Circuits (5){1,2,3,4} {1,2,3,5} {1,2,4,5} {1,3,4,5} {2,3,4,5}
Flats by rank (17)
Hyperplanes (10){1,2} {1,3} {2,3} {1,4} {2,4} {3,4} {1,5} {2,5} {3,5} {4,5}
Lines (0)
Dual (rank 2)

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

Bases: {4,5} {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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