Matroid 3.6.20

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

Basic invariants

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

Status	computed_realization_space
Dimension of realization space	0
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	1,4,5

Realization space: scheme simple core id	`r3_n4_****`
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_n4_****`
Realization space: is regular scheme	yes how determined: regularity method: `simple_core` regularity witness: `r3_n4_****`
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

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

\(T = x_{0}^{3} + x_{0}^{2} x_{1}^{2} + x_{0}^{2} x_{1} + x_{0}^{2} + x_{0} x_{1}^{2} + x_{0} x_{1} + x_{0} + x_{1}^{3} + x_{1}^{2} + x_{1}\)

Computed on the fly from the id.

{1,4,5} {2,4,5} {3,4,5} {1,4,6} {2,4,6} {3,4,6} {1,5,6} {2,5,6} {3,5,6} {4,5,6}

{1,2,3} {1,2,4} {1,3,4} {2,3,4} {1,2,5} {1,3,5} {2,3,5} {1,2,6} {1,3,6} {2,3,6}

{1,2} {1,3} {2,3} {1,4,5,6} {2,4,5,6} {3,4,5,6}

{1,2,3,4} {1,2,3,5} {4,5} {1,2,3,6} {4,6} {5,6}

Dual (rank 3)

Revlex encoding in this labeling (not canonicalized, so not linked): *******000***0000000

Bases: {2,3,6} {1,3,6} {1,2,6} {2,3,5} {1,3,5} {1,2,5} {2,3,4} {1,3,4} {1,2,4} {1,2,3}

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