Matroid 4.9.84603

Label	4.9.84603
Id	`r4_n9_0******00*00********************0******0********************0******0*******0********0**0`
Rank	4
n	9

Basic invariants

Bases	114
Circuits	78
Flats	95
Cyclic flats	14
Loops	0
Connected components	1
Automorphisms	1
Beta invariant	23
Girth	4
Simple	yes
Connected	yes
Uniform	not computed
Loopless	yes
Coloopless	yes
Paving	yes
Laminar	no
Nested	no
Self dual	no
Identically self dual	no
Series parallel	no
Transversal	no
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	no
Quaternary	no
Graphic	no

Realization space

Status	computed_realization_space
Dimension of realization space	5
Expected dimension over ℤ	not computed
Components of realization space	not computed
Free realization space	no
Principal ideal	yes
Birational type	point over Q(a)/(a^3 + 2*a - 1) how determined: birational type method: `point_field`
Birational type components	show [ { "dim": 0, "free_rank": 0, "torus_rank": 0, "core_dim": 0, "qbar_components": 3, "type": "point over Q(a)/(a^3 + 2a - 1)", "field": "Q(a)/(a^3 + 2a - 1)", "minpoly": "T^3 + 2T - 1", "degree": 3, "core_vars": [ "x1" ], "core_gens": [ "4x1^3 - 15x1^2 + 14x1 - 4" ] } ]
Good basis	not computed
Good basis v2	1,2,3,8

Geometry

Realization space: scheme simple core id	not computed
Smooth char 0	yes
Singular primes	[59]
Realization space: smooth over ℚ	not computed
Realization space: smooth over ℤ	not computed
Realization space: is regular scheme	not computed
Realization space: singular fiber primes	not computed
Realization space: singular fiber characteristics	not computed
Realization space: characteristic dimensions	not computed
Realization space: characteristic dimension unexpected	not computed
Realization space: characteristic dimension varies	not computed

Other

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

Tutte polynomial

\(T = x_{0}^{4} + 5 x_{0}^{3} + 15 x_{0}^{2} + 12 x_{0} x_{1} + 23 x_{0} + x_{1}^{5} + 4 x_{1}^{4} + 10 x_{1}^{3} + 20 x_{1}^{2} + 23 x_{1}\)

Realization space

Ring	\(\mathbb{Z}[x_{1}]\)
Defining ideal	\(\left(4 x_{1}^{3} - 15 x_{1}^{2} + 14 x_{1} - 4\right)\)
Inequations	\(12 x_{1}^{2} - 37 x_{1} + 18 \neq 0\), \(4 x_{1}^{2} - 15 x_{1} + 8 \neq 0\), \(48 x_{1}^{4} - 328 x_{1}^{3} + 747 x_{1}^{2} - 640 x_{1} + 184 \neq 0\), \(48 x_{1}^{4} - 328 x_{1}^{3} + 731 x_{1}^{2} - 588 x_{1} + 156 \neq 0\), \(x_{1} - 2 \neq 0\), \(16 x_{1}^{4} - 112 x_{1}^{3} + 259 x_{1}^{2} - 220 x_{1} + 62 \neq 0\), \(12 x_{1}^{2} - 37 x_{1} + 20 \neq 0\), \(4 x_{1}^{2} - 15 x_{1} + 6 \neq 0\), \(48 x_{1}^{4} - 328 x_{1}^{3} + 747 x_{1}^{2} - 640 x_{1} + 188 \neq 0\), \(4 x_{1}^{2} - 15 x_{1} + 10 \neq 0\), \(32 x_{1}^{6} - 272 x_{1}^{5} + 882 x_{1}^{4} - 1414 x_{1}^{3} + 1223 x_{1}^{2} - 542 x_{1} + 96 \neq 0\), \(2 \neq 0\), \(4 x_{1}^{2} - 11 x_{1} + 4 \neq 0\), \(32 x_{1}^{6} - 272 x_{1}^{5} + 882 x_{1}^{4} - 1398 x_{1}^{3} + 1171 x_{1}^{2} - 504 x_{1} + 88 \neq 0\), \(64 x_{1}^{6} - 672 x_{1}^{5} + 2708 x_{1}^{4} - 5236 x_{1}^{3} + 5023 x_{1}^{2} - 2306 x_{1} + 404 \neq 0\), \(48 x_{1}^{6} - 472 x_{1}^{5} + 1787 x_{1}^{4} - 3279 x_{1}^{3} + 3033 x_{1}^{2} - 1354 x_{1} + 232 \neq 0\), \(48 x_{1}^{4} - 304 x_{1}^{3} + 633 x_{1}^{2} - 478 x_{1} + 116 \neq 0\), \(16 x_{1}^{4} - 104 x_{1}^{3} + 205 x_{1}^{2} - 122 x_{1} + 20 \neq 0\), \(48 x_{1}^{6} - 472 x_{1}^{5} + 1803 x_{1}^{4} - 3379 x_{1}^{3} + 3247 x_{1}^{2} - 1532 x_{1} + 280 \neq 0\), \(12 x_{1}^{3} - 45 x_{1}^{2} + 40 x_{1} - 8 \neq 0\), \(16 x_{1}^{4} - 104 x_{1}^{3} + 221 x_{1}^{2} - 168 x_{1} + 40 \neq 0\), \(16 x_{1}^{4} - 104 x_{1}^{3} + 213 x_{1}^{2} - 146 x_{1} + 32 \neq 0\), \(x_{1} \neq 0\), \(32 x_{1}^{5} - 232 x_{1}^{4} + 584 x_{1}^{3} - 619 x_{1}^{2} + 288 x_{1} - 48 \neq 0\), \(32 x_{1}^{5} - 232 x_{1}^{4} + 584 x_{1}^{3} - 611 x_{1}^{2} + 276 x_{1} - 44 \neq 0\), \(12 x_{1}^{3} - 37 x_{1}^{2} + 16 x_{1} + 4 \neq 0\), \(4 x_{1}^{3} - 23 x_{1}^{2} + 44 x_{1} - 20 \neq 0\), \(4 x_{1}^{3} - 23 x_{1}^{2} + 40 x_{1} - 16 \neq 0\), \(48 x_{1}^{4} - 328 x_{1}^{3} + 739 x_{1}^{2} - 610 x_{1} + 164 \neq 0\), \(48 x_{1}^{4} - 328 x_{1}^{3} + 723 x_{1}^{2} - 566 x_{1} + 140 \neq 0\), \(48 x_{1}^{4} - 328 x_{1}^{3} + 771 x_{1}^{2} - 714 x_{1} + 220 \neq 0\), \(4 x_{1}^{3} - 23 x_{1}^{2} + 36 x_{1} - 8 \neq 0\), \(12 x_{1}^{2} - 37 x_{1} + 16 \neq 0\), \(32 x_{1}^{6} - 272 x_{1}^{5} + 898 x_{1}^{4} - 1506 x_{1}^{3} + 1403 x_{1}^{2} - 682 x_{1} + 132 \neq 0\), \(16 x_{1}^{4} - 128 x_{1}^{3} + 311 x_{1}^{2} - 258 x_{1} + 68 \neq 0\), \(16 x_{1}^{4} - 104 x_{1}^{3} + 221 x_{1}^{2} - 174 x_{1} + 44 \neq 0\), \(32 x_{1}^{6} - 272 x_{1}^{5} + 866 x_{1}^{4} - 1274 x_{1}^{3} + 871 x_{1}^{2} - 250 x_{1} + 20 \neq 0\), \(4 x_{1}^{2} - 7 x_{1} + 4 \neq 0\), \(8 x_{1}^{4} - 38 x_{1}^{3} + 58 x_{1}^{2} - 35 x_{1} + 8 \neq 0\), \(4 x_{1}^{4} - 19 x_{1}^{3} + 31 x_{1}^{2} - 23 x_{1} + 6 \neq 0\), \(4 x_{1}^{2} - 13 x_{1} + 8 \neq 0\), \(16 x_{1}^{4} - 104 x_{1}^{3} + 237 x_{1}^{2} - 214 x_{1} + 60 \neq 0\), \(48 x_{1}^{4} - 280 x_{1}^{3} + 559 x_{1}^{2} - 442 x_{1} + 116 \neq 0\), \(12 x_{1}^{3} - 45 x_{1}^{2} + 38 x_{1} - 8 \neq 0\)
Realization matrix	\(\begin{pmatrix}1 & 0 & 0 & 1 & 4 & 2 & 0 & 0 & 2 \\ 0 & 1 & 0 & 1 & -4 x_{1}^{2} + 15 x_{1} - 10 & 0 & 2 & 0 & -4 x_{1}^{2} + 15 x_{1} - 8 \\ 0 & 0 & 1 & 1 & 12 x_{1}^{2} - 37 x_{1} + 20 & 4 x_{1}^{2} - 11 x_{1} + 6 & -12 x_{1}^{2} + 37 x_{1} - 18 & 0 & 0 \\ 0 & 0 & 0 & 0 & 4 & 2 x_{1} & 8 x_{1}^{2} - 22 x_{1} + 8 & 1 & 2 x_{1}\end{pmatrix}\)

Combinatorics

Computed on the fly from the id.

Bases (114)

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

Non-bases (12)

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

Circuits (78)

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

Flats by rank (95)

rank 0 (1): {}
rank 1 (9): {1} {2} {3} {4} {5} {6} {7} {8} {9}
rank 2 (36): {1,2} {1,3} {2,3} {1,4} {2,4} {3,4} {1,5} {2,5} {3,5} {4,5} {1,6} {2,6} {3,6} {4,6} {5,6} {1,7} {2,7} {3,7} {4,7} {5,7} {6,7} {1,8} {2,8} {3,8} {4,8} {5,8} {6,8} {7,8} {1,9} {2,9} {3,9} {4,9} {5,9} {6,9} {7,9} {8,9}
rank 3 (48): {1,2,3,4} {2,3,5} {1,4,5} {1,4,6} {2,4,6} {1,2,5,6} {3,4,5,6} {1,2,7} {3,4,7} {1,3,5,7} {2,4,5,7} {1,6,7} {2,6,7} {3,6,7} {4,6,7} {5,6,7} {1,4,8} {2,4,8} {3,4,8} {1,5,8} {2,5,8} {3,5,8} {4,5,8} {2,6,8} {1,3,6,8} {5,6,8} {1,7,8} {2,3,7,8} {4,7,8} {6,7,8} {1,3,9} {2,4,9} {3,4,9} {1,5,9} {2,5,9} {3,5,9} {4,5,9} {1,6,9} {2,3,6,9} {5,6,9} {2,7,9} {3,7,9} {1,4,7,9} {6,7,9} {1,2,8,9} {3,8,9} {4,6,8,9} {5,7,8,9}
rank 4 (1): {1,2,3,4,5,6,7,8,9}

Hyperplanes (48)

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

Dual (rank 5)

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

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

Loops: none. Parallel classes of size > 1: none.

Downloads

Record as JSON
OSCAR: matroid_from_revlex_basis_encoding("0********0****0*****0**0**********************0**********0************************0**********0***********0************0*****0*", 4, 9)