Matroid 3.10.9161

Label	3.10.9161
Id	`r3_n10_000000000000000000000000000**00000000000000000000000000000000000**0000000000`
Rank	3
n	10

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

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

Basic invariants

Bases	48
Circuits	29
Flats	13
Cyclic flats	12
Loops	1
Connected components	2
Automorphisms	128
Beta invariant	0
Girth	1
Simple	no
Connected	no
Uniform	not computed
Loopless	no
Coloopless	yes
Paving	no
Laminar	no
Nested	no
Self dual	no
Identically self dual	no
Series parallel	no
Transversal	no
Supersolvable	yes
Divisionally free	yes
Orientable	yes
Three lines	unknown

Representability

Characteristic set	characteristic 0 and all primes `[0]`
Realizable	yes
Realizable char0	yes
Regular	yes
Binary	yes
Ternary	yes
Quaternary	yes
Graphic	yes

Realization space

Status	computed_realization_space
Dimension of realization space	2
Expected dimension over ℤ	1
Components of realization space	1
Free realization space	yes
Principal ideal	yes
Birational type	rational 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 basis	unknown
Good basis v2	2,6,7

Geometry

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

Other

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

Tutte polynomial

\(T = x_{0}^{3} x_{1} + 4 x_{0}^{2} x_{1}^{2} + 2 x_{0}^{2} x_{1} + 2 x_{0} x_{1}^{4} + 8 x_{0} x_{1}^{3} + 6 x_{0} x_{1}^{2} + x_{0} x_{1} + x_{1}^{7} + 3 x_{1}^{6} + 6 x_{1}^{5} + 8 x_{1}^{4} + 5 x_{1}^{3} + x_{1}^{2}\)

Realization space

Ring	\(\mathbb{Z}\)
Defining ideal	\(\left(0\right)\)
Inequations	none
Realization matrix	\(\begin{pmatrix}0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1\end{pmatrix}\)

Combinatorics

Computed on the fly from the id.

Bases (48)

{2,4,7} {3,4,7} {2,5,7} {3,5,7} {2,6,7} {3,6,7} {4,6,7} {5,6,7} {2,4,8} {3,4,8} {2,5,8} {3,5,8} {2,6,8} {3,6,8} {4,6,8} {5,6,8} {2,4,9} {3,4,9} {2,5,9} {3,5,9} {2,6,9} {3,6,9} {4,6,9} {5,6,9} {2,7,9} {3,7,9} {4,7,9} {5,7,9} {2,8,9} {3,8,9} {4,8,9} {5,8,9} {2,4,10} {3,4,10} {2,5,10} {3,5,10} {2,6,10} {3,6,10} {4,6,10} {5,6,10} {2,7,10} {3,7,10} {4,7,10} {5,7,10} {2,8,10} {3,8,10} {4,8,10} {5,8,10}

Non-bases (72)

{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} {1,2,6} {1,3,6} {2,3,6} {1,4,6} {2,4,6} {3,4,6} {1,5,6} {2,5,6} {3,5,6} {4,5,6} {1,2,7} {1,3,7} {2,3,7} {1,4,7} {1,5,7} {4,5,7} {1,6,7} {1,2,8} {1,3,8} {2,3,8} {1,4,8} {1,5,8} {4,5,8} {1,6,8} {1,7,8} {2,7,8} {3,7,8} {4,7,8} {5,7,8} {6,7,8} {1,2,9} {1,3,9} {2,3,9} {1,4,9} {1,5,9} {4,5,9} {1,6,9} {1,7,9} {6,7,9} {1,8,9} {6,8,9} {7,8,9} {1,2,10} {1,3,10} {2,3,10} {1,4,10} {1,5,10} {4,5,10} {1,6,10} {1,7,10} {6,7,10} {1,8,10} {6,8,10} {7,8,10} {1,9,10} {2,9,10} {3,9,10} {4,9,10} {5,9,10} {6,9,10} {7,9,10} {8,9,10}

Circuits (29)

{1} {2,3} {4,5} {2,4,6} {3,4,6} {2,5,6} {3,5,6} {7,8} {2,4,7,9} {3,4,7,9} {2,5,7,9} {3,5,7,9} {6,7,9} {2,4,8,9} {3,4,8,9} {2,5,8,9} {3,5,8,9} {6,8,9} {2,4,7,10} {3,4,7,10} {2,5,7,10} {3,5,7,10} {6,7,10} {2,4,8,10} {3,4,8,10} {2,5,8,10} {3,5,8,10} {6,8,10} {9,10}

Flats by rank (13)

rank 0 (1): {1}
rank 1 (5): {1,2,3} {1,4,5} {1,6} {1,7,8} {1,9,10}
rank 2 (6): {1,2,3,4,5,6} {1,2,3,7,8} {1,4,5,7,8} {1,2,3,9,10} {1,4,5,9,10} {1,6,7,8,9,10}
rank 3 (1): {1,2,3,4,5,6,7,8,9,10}

Hyperplanes (6)

{1,2,3,4,5,6} {1,2,3,7,8} {1,4,5,7,8} {1,2,3,9,10} {1,4,5,9,10} {1,6,7,8,9,10}

Lines (2)

{1,2,3,4,5,6} {1,6,7,8,9,10}

Dual (rank 7)

Revlex encoding in this labeling (not canonicalized, so not linked): 0000000000****00****0****00**0**000000****00****0****00**0**0000000000****00**0**0000****00**0**000000000000000000000000

Bases: {1,3,5,6,8,9,10} {1,2,5,6,8,9,10} {1,3,4,6,8,9,10} {1,2,4,6,8,9,10} {1,3,4,5,8,9,10} {1,2,4,5,8,9,10} {1,2,3,5,8,9,10} {1,2,3,4,8,9,10} {1,3,5,6,7,9,10} {1,2,5,6,7,9,10} {1,3,4,6,7,9,10} {1,2,4,6,7,9,10} {1,3,4,5,7,9,10} {1,2,4,5,7,9,10} {1,2,3,5,7,9,10} {1,2,3,4,7,9,10} {1,3,5,6,7,8,10} {1,2,5,6,7,8,10} {1,3,4,6,7,8,10} {1,2,4,6,7,8,10} {1,3,4,5,7,8,10} {1,2,4,5,7,8,10} {1,2,3,5,7,8,10} {1,2,3,4,7,8,10} {1,3,4,5,6,8,10} {1,2,4,5,6,8,10} {1,2,3,5,6,8,10} {1,2,3,4,6,8,10} {1,3,4,5,6,7,10} {1,2,4,5,6,7,10} {1,2,3,5,6,7,10} {1,2,3,4,6,7,10} {1,3,5,6,7,8,9} {1,2,5,6,7,8,9} {1,3,4,6,7,8,9} {1,2,4,6,7,8,9} {1,3,4,5,7,8,9} {1,2,4,5,7,8,9} {1,2,3,5,7,8,9} {1,2,3,4,7,8,9} {1,3,4,5,6,8,9} {1,2,4,5,6,8,9} {1,2,3,5,6,8,9} {1,2,3,4,6,8,9} {1,3,4,5,6,7,9} {1,2,4,5,6,7,9} {1,2,3,5,6,7,9} {1,2,3,4,6,7,9}

Loops: 1. Parallel classes of size > 1: {2,3} {4,5} {7,8} {9,10}.

Downloads

Record as JSON
OSCAR: matroid_from_revlex_basis_encoding("000000000000000000000000**0**00****0000**0**00****0000000000**0**00****0****00****000000**0**00****0****00****0000000000", 3, 10)