Matroid database

Matroid 3.5.1

Label	3.5.1
Id	`r3_n5_0*********`
Rank	3
n	5

Affine diagram (real realization).

Basic invariants

Bases	9
Circuits	4
Flats	15
Cyclic flats	not computed
Loops	0
Connected components	1
Automorphisms	not computed
Beta invariant	not computed
Girth	3
Simple	yes
Connected	yes
Uniform	no
Loopless	yes
Coloopless	yes
Paving	yes
Laminar	not computed
Nested	not computed
Self dual	not computed
Identically self dual	not computed
Series parallel	not computed
Transversal	not computed
Supersolvable	no
Divisionally free	no
Orientable	yes
Three lines	not computed

Representability

Characteristic set	characteristic 0 and all primes `[0]`
Realizable	yes
Realizable char0	yes
Regular	no
Binary	no
Ternary	yes
Quaternary	yes
Graphic	no

Realization space

Status	computed_realization_space
Dimension of realization space	1
Expected dimension over ℤ	2
Components of realization space	1
Free realization space	yes
Principal ideal	yes
Birational type	rational 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 basis	not computed
Good basis v2	1,2,4

Geometry

Realization space: scheme simple core id	`r3_n5_0*********`
Smooth char 0	not computed
Singular primes	not computed
Realization space: smooth over ℚ	not computed
Realization space: smooth over ℤ	yes how determined: smoothness method: `three_lines_deletion` smoothness witness: `1=>r3_n4_****`
Realization space: is regular scheme	yes 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 characteristics	not computed
Realization space: characteristic dimensions	show [ { "p": 0, "d": 1 } ] how determined: characteristic dimensions method: `cofinite_smooth_equidimensional`
Realization space: characteristic dimension unexpected	no
Realization space: characteristic dimension varies	no

Other

Char poly splits	false
Simplicial arrangement	no
Realization space: n qbar components	1

Tutte polynomial

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

Realization space

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

Combinatorics

Computed on the fly from the id.

{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}

{1,2,3}

{1,2,3} {1,2,4,5} {1,3,4,5} {2,3,4,5}

Flats by rank (15)

rank 0 (1): {}
rank 1 (5): {1} {2} {3} {4} {5}
rank 2 (8): {1,2,3} {1,4} {2,4} {3,4} {1,5} {2,5} {3,5} {4,5}
rank 3 (1): {1,2,3,4,5}

Hyperplanes (8)

{1,2,3} {1,4} {2,4} {3,4} {1,5} {2,5} {3,5} {4,5}

{1,2,3}

Dual (rank 2)

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

Bases: {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.

Downloads

Record as JSON
OSCAR: matroid_from_revlex_basis_encoding("0*********", 3, 5)