Matroid 3.12.31809519

Label	3.12.31809519
Id	`r3_n12_000000000000000000000**0*****00*****0000*****0000000*******00**0***00**0*********0**000*00***00**********000*****000****00*****0`
Rank	3
n	12

Affine diagram.

Basic invariants

Bases	162
Circuits	166
Flats	24
Cyclic flats	16
Loops	0
Connected components	1
Automorphisms	48
Beta invariant	10
Girth	2
Simple	no
Connected	yes
Uniform	not computed
Loopless	yes
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	no
Three lines	no

Representability

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

Realization space

Status	computed_realization_space
Dimension of realization space	3
Expected dimension over ℤ	-2
Components of realization space	not computed
Free realization space	no
Principal ideal	yes
Birational type	not computed
Birational type components	not computed
Good basis	not computed
Good basis v2	1,3,12

Geometry

Realization space: scheme simple core id	not computed
Smooth char 0	not computed
Singular primes	not computed
Realization space: smooth over ℚ	yes
Realization space: smooth over ℤ	no how determined: smoothness method: `simple_core` smoothness witness: `r3_n10_0000**********0*****000***00*0******000**0************00***00***0****0`
Realization space: is regular scheme	not computed
Realization space: singular fiber primes	not computed
Realization space: singular fiber characteristics	none how determined: singular fiber characteristics method: `complete_simple_core`
Realization space: characteristic dimensions	show [ { "p": 3, "d": 0 } ] how determined: characteristic dimensions method: `simple_core`
Realization space: characteristic dimension unexpected	yes
Realization space: characteristic dimension varies	no

Other

Char poly splits	true
Simplicial arrangement	no
Realization space: n qbar components	not computed

Tutte polynomial

\(T = x_{0}^{3} + 2 x_{0}^{2} x_{1} + 7 x_{0}^{2} + x_{0} x_{1}^{4} + 4 x_{0} x_{1}^{3} + 10 x_{0} x_{1}^{2} + 18 x_{0} x_{1} + 10 x_{0} + x_{1}^{9} + 3 x_{1}^{8} + 6 x_{1}^{7} + 10 x_{1}^{6} + 15 x_{1}^{5} + 20 x_{1}^{4} + 23 x_{1}^{3} + 21 x_{1}^{2} + 10 x_{1}\)

Realization space

Ring	\(\mathbb{Z}\)
Defining ideal	\(\left(3\right)\)
Inequations	none
Realization matrix	\(\begin{pmatrix}1 & 1 & 0 & 0 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 & 1 & 2 & 2 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 & 2 & 1 & 2 & 1\end{pmatrix}\)

Combinatorics

Computed on the fly from the id.

Bases (162)

{1,3,7} {2,3,7} {1,4,7} {2,4,7} {1,5,7} {2,5,7} {3,5,7} {4,5,7} {1,6,7} {2,6,7} {3,6,7} {4,6,7} {5,6,7} {1,3,8} {2,3,8} {1,4,8} {2,4,8} {1,5,8} {2,5,8} {3,5,8} {4,5,8} {1,6,8} {2,6,8} {3,6,8} {4,6,8} {5,6,8} {3,7,8} {4,7,8} {5,7,8} {6,7,8} {1,3,9} {2,3,9} {1,4,9} {2,4,9} {1,5,9} {2,5,9} {3,5,9} {4,5,9} {1,6,9} {2,6,9} {3,6,9} {4,6,9} {5,6,9} {3,7,9} {4,7,9} {5,7,9} {6,7,9} {3,8,9} {4,8,9} {5,8,9} {6,8,9} {1,3,10} {2,3,10} {1,4,10} {2,4,10} {1,5,10} {2,5,10} {3,5,10} {4,5,10} {1,6,10} {2,6,10} {3,6,10} {4,6,10} {5,6,10} {1,7,10} {2,7,10} {5,7,10} {6,7,10} {1,8,10} {2,8,10} {3,8,10} {4,8,10} {6,8,10} {7,8,10} {1,9,10} {2,9,10} {3,9,10} {4,9,10} {5,9,10} {7,9,10} {8,9,10} {1,3,11} {2,3,11} {1,4,11} {2,4,11} {1,5,11} {2,5,11} {3,5,11} {4,5,11} {1,6,11} {2,6,11} {3,6,11} {4,6,11} {5,6,11} {1,7,11} {2,7,11} {3,7,11} {4,7,11} {6,7,11} {1,8,11} {2,8,11} {3,8,11} {4,8,11} {5,8,11} {7,8,11} {1,9,11} {2,9,11} {5,9,11} {6,9,11} {7,9,11} {8,9,11} {3,10,11} {4,10,11} {5,10,11} {6,10,11} {7,10,11} {8,10,11} {9,10,11} {1,3,12} {2,3,12} {1,4,12} {2,4,12} {1,5,12} {2,5,12} {3,5,12} {4,5,12} {1,6,12} {2,6,12} {3,6,12} {4,6,12} {5,6,12} {1,7,12} {2,7,12} {3,7,12} {4,7,12} {5,7,12} {1,8,12} {2,8,12} {5,8,12} {6,8,12} {7,8,12} {1,9,12} {2,9,12} {3,9,12} {4,9,12} {6,9,12} {7,9,12} {8,9,12} {3,10,12} {4,10,12} {5,10,12} {6,10,12} {7,10,12} {8,10,12} {9,10,12} {3,11,12} {4,11,12} {5,11,12} {6,11,12} {7,11,12} {8,11,12} {9,11,12}

Non-bases (58)

{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} {3,4,7} {1,2,8} {3,4,8} {1,7,8} {2,7,8} {1,2,9} {3,4,9} {1,7,9} {2,7,9} {1,8,9} {2,8,9} {7,8,9} {1,2,10} {3,4,10} {3,7,10} {4,7,10} {5,8,10} {6,9,10} {1,2,11} {3,4,11} {5,7,11} {6,8,11} {3,9,11} {4,9,11} {1,10,11} {2,10,11} {1,2,12} {3,4,12} {6,7,12} {3,8,12} {4,8,12} {5,9,12} {1,10,12} {2,10,12} {1,11,12} {2,11,12} {10,11,12}

Circuits (166)

{1,2} {3,4} {1,3,5} {2,3,5} {1,4,5} {2,4,5} {1,3,6} {2,3,6} {1,4,6} {2,4,6} {1,5,6} {2,5,6} {3,5,6} {4,5,6} {1,7,8} {2,7,8} {3,5,7,8} {4,5,7,8} {3,6,7,8} {4,6,7,8} {5,6,7,8} {1,7,9} {2,7,9} {3,5,7,9} {4,5,7,9} {3,6,7,9} {4,6,7,9} {5,6,7,9} {1,8,9} {2,8,9} {3,5,8,9} {4,5,8,9} {3,6,8,9} {4,6,8,9} {5,6,8,9} {7,8,9} {3,7,10} {4,7,10} {1,5,7,10} {2,5,7,10} {1,6,7,10} {2,6,7,10} {5,6,7,10} {1,3,8,10} {2,3,8,10} {1,4,8,10} {2,4,8,10} {5,8,10} {1,6,8,10} {2,6,8,10} {3,6,8,10} {4,6,8,10} {6,7,8,10} {1,3,9,10} {2,3,9,10} {1,4,9,10} {2,4,9,10} {1,5,9,10} {2,5,9,10} {3,5,9,10} {4,5,9,10} {6,9,10} {5,7,9,10} {3,8,9,10} {4,8,9,10} {1,3,7,11} {2,3,7,11} {1,4,7,11} {2,4,7,11} {5,7,11} {1,6,7,11} {2,6,7,11} {3,6,7,11} {4,6,7,11} {1,3,8,11} {2,3,8,11} {1,4,8,11} {2,4,8,11} {1,5,8,11} {2,5,8,11} {3,5,8,11} {4,5,8,11} {6,8,11} {3,7,8,11} {4,7,8,11} {3,9,11} {4,9,11} {1,5,9,11} {2,5,9,11} {1,6,9,11} {2,6,9,11} {5,6,9,11} {6,7,9,11} {5,8,9,11} {1,10,11} {2,10,11} {3,5,10,11} {4,5,10,11} {3,6,10,11} {4,6,10,11} {5,6,10,11} {6,7,10,11} {3,8,10,11} {4,8,10,11} {7,8,10,11} {5,9,10,11} {7,9,10,11} {8,9,10,11} {1,3,7,12} {2,3,7,12} {1,4,7,12} {2,4,7,12} {1,5,7,12} {2,5,7,12} {3,5,7,12} {4,5,7,12} {6,7,12} {3,8,12} {4,8,12} {1,5,8,12} {2,5,8,12} {1,6,8,12} {2,6,8,12} {5,6,8,12} {5,7,8,12} {1,3,9,12} {2,3,9,12} {1,4,9,12} {2,4,9,12} {5,9,12} {1,6,9,12} {2,6,9,12} {3,6,9,12} {4,6,9,12} {3,7,9,12} {4,7,9,12} {6,8,9,12} {1,10,12} {2,10,12} {3,5,10,12} {4,5,10,12} {3,6,10,12} {4,6,10,12} {5,6,10,12} {5,7,10,12} {6,8,10,12} {7,8,10,12} {3,9,10,12} {4,9,10,12} {7,9,10,12} {8,9,10,12} {1,11,12} {2,11,12} {3,5,11,12} {4,5,11,12} {3,6,11,12} {4,6,11,12} {5,6,11,12} {3,7,11,12} {4,7,11,12} {5,8,11,12} {7,8,11,12} {6,9,11,12} {7,9,11,12} {8,9,11,12} {10,11,12}

Flats by rank (24)

rank 0 (1): {}
rank 1 (10): {1,2} {3,4} {5} {6} {7} {8} {9} {10} {11} {12}
rank 2 (12): {1,2,3,4,5,6} {1,2,7,8,9} {3,4,7,10} {5,8,10} {6,9,10} {5,7,11} {6,8,11} {3,4,9,11} {6,7,12} {3,4,8,12} {5,9,12} {1,2,10,11,12}
rank 3 (1): {1,2,3,4,5,6,7,8,9,10,11,12}

Hyperplanes (12)

{1,2,3,4,5,6} {1,2,7,8,9} {3,4,7,10} {5,8,10} {6,9,10} {5,7,11} {6,8,11} {3,4,9,11} {6,7,12} {3,4,8,12} {5,9,12} {1,2,10,11,12}

Lines (12)

{1,2,3,4,5,6} {1,2,7,8,9} {3,4,7,10} {5,8,10} {6,9,10} {5,7,11} {6,8,11} {3,4,9,11} {6,7,12} {3,4,8,12} {5,9,12} {1,2,10,11,12}

Dual (rank 9)

Revlex encoding in this labeling (not canonicalized, so not linked): 0*******00*******00***0*******00**0**************0****0*******00****00***0******0*************0****0**0*******0******00***********0****00****00****00*********0****0****00*********0****0*********0****000000000000000000000

Bases: {2,4,5,6,8,9,10,11,12} {1,4,5,6,8,9,10,11,12} {2,3,5,6,8,9,10,11,12} {1,3,5,6,8,9,10,11,12} {2,3,4,6,8,9,10,11,12} {1,3,4,6,8,9,10,11,12} {1,2,4,6,8,9,10,11,12} {1,2,3,6,8,9,10,11,12} {2,3,4,5,8,9,10,11,12} {1,3,4,5,8,9,10,11,12} {1,2,4,5,8,9,10,11,12} {1,2,3,5,8,9,10,11,12} {1,2,3,4,8,9,10,11,12} {2,4,5,6,7,9,10,11,12} {1,4,5,6,7,9,10,11,12} {2,3,5,6,7,9,10,11,12} {1,3,5,6,7,9,10,11,12} {2,3,4,6,7,9,10,11,12} {1,3,4,6,7,9,10,11,12} {1,2,4,6,7,9,10,11,12} {1,2,3,6,7,9,10,11,12} {2,3,4,5,7,9,10,11,12} {1,3,4,5,7,9,10,11,12} {1,2,4,5,7,9,10,11,12} {1,2,3,5,7,9,10,11,12} {1,2,3,4,7,9,10,11,12} {1,2,4,5,6,9,10,11,12} {1,2,3,5,6,9,10,11,12} {1,2,3,4,6,9,10,11,12} {1,2,3,4,5,9,10,11,12} {2,4,5,6,7,8,10,11,12} {1,4,5,6,7,8,10,11,12} {2,3,5,6,7,8,10,11,12} {1,3,5,6,7,8,10,11,12} {2,3,4,6,7,8,10,11,12} {1,3,4,6,7,8,10,11,12} {1,2,4,6,7,8,10,11,12} {1,2,3,6,7,8,10,11,12} {2,3,4,5,7,8,10,11,12} {1,3,4,5,7,8,10,11,12} {1,2,4,5,7,8,10,11,12} {1,2,3,5,7,8,10,11,12} {1,2,3,4,7,8,10,11,12} {1,2,4,5,6,8,10,11,12} {1,2,3,5,6,8,10,11,12} {1,2,3,4,6,8,10,11,12} {1,2,3,4,5,8,10,11,12} {1,2,4,5,6,7,10,11,12} {1,2,3,5,6,7,10,11,12} {1,2,3,4,6,7,10,11,12} {1,2,3,4,5,7,10,11,12} {2,4,5,6,7,8,9,11,12} {1,4,5,6,7,8,9,11,12} {2,3,5,6,7,8,9,11,12} {1,3,5,6,7,8,9,11,12} {2,3,4,6,7,8,9,11,12} {1,3,4,6,7,8,9,11,12} {1,2,4,6,7,8,9,11,12} {1,2,3,6,7,8,9,11,12} {2,3,4,5,7,8,9,11,12} {1,3,4,5,7,8,9,11,12} {1,2,4,5,7,8,9,11,12} {1,2,3,5,7,8,9,11,12} {1,2,3,4,7,8,9,11,12} {2,3,4,5,6,8,9,11,12} {1,3,4,5,6,8,9,11,12} {1,2,3,4,6,8,9,11,12} {1,2,3,4,5,8,9,11,12} {2,3,4,5,6,7,9,11,12} {1,3,4,5,6,7,9,11,12} {1,2,4,5,6,7,9,11,12} {1,2,3,5,6,7,9,11,12} {1,2,3,4,5,7,9,11,12} {1,2,3,4,5,6,9,11,12} {2,3,4,5,6,7,8,11,12} {1,3,4,5,6,7,8,11,12} {1,2,4,5,6,7,8,11,12} {1,2,3,5,6,7,8,11,12} {1,2,3,4,6,7,8,11,12} {1,2,3,4,5,6,8,11,12} {1,2,3,4,5,6,7,11,12} {2,4,5,6,7,8,9,10,12} {1,4,5,6,7,8,9,10,12} {2,3,5,6,7,8,9,10,12} {1,3,5,6,7,8,9,10,12} {2,3,4,6,7,8,9,10,12} {1,3,4,6,7,8,9,10,12} {1,2,4,6,7,8,9,10,12} {1,2,3,6,7,8,9,10,12} {2,3,4,5,7,8,9,10,12} {1,3,4,5,7,8,9,10,12} {1,2,4,5,7,8,9,10,12} {1,2,3,5,7,8,9,10,12} {1,2,3,4,7,8,9,10,12} {2,3,4,5,6,8,9,10,12} {1,3,4,5,6,8,9,10,12} {1,2,4,5,6,8,9,10,12} {1,2,3,5,6,8,9,10,12} {1,2,3,4,5,8,9,10,12} {2,3,4,5,6,7,9,10,12} {1,3,4,5,6,7,9,10,12} {1,2,4,5,6,7,9,10,12} {1,2,3,5,6,7,9,10,12} {1,2,3,4,6,7,9,10,12} {1,2,3,4,5,6,9,10,12} {2,3,4,5,6,7,8,10,12} {1,3,4,5,6,7,8,10,12} {1,2,3,4,6,7,8,10,12} {1,2,3,4,5,7,8,10,12} {1,2,3,4,5,6,8,10,12} {1,2,3,4,5,6,7,10,12} {1,2,4,5,6,7,8,9,12} {1,2,3,5,6,7,8,9,12} {1,2,3,4,6,7,8,9,12} {1,2,3,4,5,7,8,9,12} {1,2,3,4,5,6,8,9,12} {1,2,3,4,5,6,7,9,12} {1,2,3,4,5,6,7,8,12} {2,4,5,6,7,8,9,10,11} {1,4,5,6,7,8,9,10,11} {2,3,5,6,7,8,9,10,11} {1,3,5,6,7,8,9,10,11} {2,3,4,6,7,8,9,10,11} {1,3,4,6,7,8,9,10,11} {1,2,4,6,7,8,9,10,11} {1,2,3,6,7,8,9,10,11} {2,3,4,5,7,8,9,10,11} {1,3,4,5,7,8,9,10,11} {1,2,4,5,7,8,9,10,11} {1,2,3,5,7,8,9,10,11} {1,2,3,4,7,8,9,10,11} {2,3,4,5,6,8,9,10,11} {1,3,4,5,6,8,9,10,11} {1,2,4,5,6,8,9,10,11} {1,2,3,5,6,8,9,10,11} {1,2,3,4,6,8,9,10,11} {2,3,4,5,6,7,9,10,11} {1,3,4,5,6,7,9,10,11} {1,2,3,4,6,7,9,10,11} {1,2,3,4,5,7,9,10,11} {1,2,3,4,5,6,9,10,11} {2,3,4,5,6,7,8,10,11} {1,3,4,5,6,7,8,10,11} {1,2,4,5,6,7,8,10,11} {1,2,3,5,6,7,8,10,11} {1,2,3,4,5,7,8,10,11} {1,2,3,4,5,6,8,10,11} {1,2,3,4,5,6,7,10,11} {1,2,4,5,6,7,8,9,11} {1,2,3,5,6,7,8,9,11} {1,2,3,4,6,7,8,9,11} {1,2,3,4,5,7,8,9,11} {1,2,3,4,5,6,8,9,11} {1,2,3,4,5,6,7,9,11} {1,2,3,4,5,6,7,8,11} {1,2,4,5,6,7,8,9,10} {1,2,3,5,6,7,8,9,10} {1,2,3,4,6,7,8,9,10} {1,2,3,4,5,7,8,9,10} {1,2,3,4,5,6,8,9,10} {1,2,3,4,5,6,7,9,10} {1,2,3,4,5,6,7,8,10}

Loops: none. Parallel classes of size > 1: {1,2} {3,4}.

Downloads

Record as JSON
OSCAR: matroid_from_revlex_basis_encoding("000000000000000000000****0*********0****0*********00****0****0*********00****00****00****0***********00******0*******0**0****0*************0******0***00****00*******0****0**************0**00*******0***00*******00*******0", 3, 12)