Matroid 3.12.31899085
| Label | 3.12.31899085 |
|---|---|
| Id | r3_n12_000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000*000000**000000***000000**** |
| Rank | 3 |
| n | 12 |
Affine diagram (real realization). Loops (not drawn): 1, 2, 3, 4, 5, 6.
Basic invariants
Representability
| Characteristic set | characteristic 0 and all primes except 2 [0,2] |
|---|---|
| Realizable | yes |
| Realizable char0 | yes |
| Regular | no |
| Binary | no |
| Ternary | no |
| Quaternary | yes |
| Graphic | no |
Realization space
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 | yes how determined: smoothness method: audited_free_chart |
| 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: smooth_over_ZZ |
| Realization space: characteristic dimensions | show[
{
"p": 0,
"d": 2
}
]how determined: characteristic dimensions method: audited_free_chart |
| Realization space: characteristic dimension unexpected | no |
| Realization space: characteristic dimension varies | no |
Other
| Char poly splits | true |
|---|---|
| Simplicial arrangement | yes |
| Realization space: n qbar components | not computed |
Tutte polynomial
\(T = x_{0}^{3} x_{1}^{6} + 3 x_{0}^{2} x_{1}^{6} + x_{0} x_{1}^{9} + 2 x_{0} x_{1}^{8} + 3 x_{0} x_{1}^{7}\)
Realization space
| Ring | \(\mathbb{Z}[x_{1}, x_{2}]\) |
|---|---|
| Defining ideal | \(\left(0\right)\) |
| Inequations | \(x_{2} \neq 0\), \(x_{2} - 1 \neq 0\), \(x_{1} - x_{2} \neq 0\), \(x_{1} \neq 0\), \(x_{1} - 1 \neq 0\) |
| Realization matrix | \(\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & x_{1} & x_{2} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}\) |
Combinatorics
Computed on the fly from the id.
Bases (10)
{7,8,12} {7,9,12} {8,9,12} {7,10,12} {8,10,12} {9,10,12} {7,11,12} {8,11,12} {9,11,12} {10,11,12}Non-bases (210)
{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} {2,4,7} {3,4,7} {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} {1,2,8} {1,3,8} {2,3,8} {1,4,8} {2,4,8} {3,4,8} {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} {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} {2,4,9} {3,4,9} {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} {1,7,9} {2,7,9} {3,7,9} {4,7,9} {5,7,9} {6,7,9} {1,8,9} {2,8,9} {3,8,9} {4,8,9} {5,8,9} {6,8,9} {7,8,9} {1,2,10} {1,3,10} {2,3,10} {1,4,10} {2,4,10} {3,4,10} {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} {3,7,10} {4,7,10} {5,7,10} {6,7,10} {1,8,10} {2,8,10} {3,8,10} {4,8,10} {5,8,10} {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} {1,2,11} {1,3,11} {2,3,11} {1,4,11} {2,4,11} {3,4,11} {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} {5,7,11} {6,7,11} {1,8,11} {2,8,11} {3,8,11} {4,8,11} {5,8,11} {6,8,11} {7,8,11} {1,9,11} {2,9,11} {3,9,11} {4,9,11} {5,9,11} {6,9,11} {7,9,11} {8,9,11} {1,10,11} {2,10,11} {3,10,11} {4,10,11} {5,10,11} {6,10,11} {7,10,11} {8,10,11} {9,10,11} {1,2,12} {1,3,12} {2,3,12} {1,4,12} {2,4,12} {3,4,12} {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} {6,7,12} {1,8,12} {2,8,12} {3,8,12} {4,8,12} {5,8,12} {6,8,12} {1,9,12} {2,9,12} {3,9,12} {4,9,12} {5,9,12} {6,9,12} {1,10,12} {2,10,12} {3,10,12} {4,10,12} {5,10,12} {6,10,12} {1,11,12} {2,11,12} {3,11,12} {4,11,12} {5,11,12} {6,11,12}Circuits (16)
{1} {2} {3} {4} {5} {6} {7,8,9} {7,8,10} {7,9,10} {8,9,10} {7,8,11} {7,9,11} {8,9,11} {7,10,11} {8,10,11} {9,10,11}Flats by rank (14)
- rank 0 (1): {1,2,3,4,5,6}
- rank 1 (6): {1,2,3,4,5,6,7} {1,2,3,4,5,6,8} {1,2,3,4,5,6,9} {1,2,3,4,5,6,10} {1,2,3,4,5,6,11} {1,2,3,4,5,6,12}
- rank 2 (6): {1,2,3,4,5,6,7,8,9,10,11} {1,2,3,4,5,6,7,12} {1,2,3,4,5,6,8,12} {1,2,3,4,5,6,9,12} {1,2,3,4,5,6,10,12} {1,2,3,4,5,6,11,12}
- rank 3 (1): {1,2,3,4,5,6,7,8,9,10,11,12}
Hyperplanes (6)
{1,2,3,4,5,6,7,8,9,10,11} {1,2,3,4,5,6,7,12} {1,2,3,4,5,6,8,12} {1,2,3,4,5,6,9,12} {1,2,3,4,5,6,10,12} {1,2,3,4,5,6,11,12}Lines (1)
{1,2,3,4,5,6,7,8,9,10,11}Dual (rank 9)
Revlex encoding in this labeling (not canonicalized, so not linked): ****000000***000000**000000*000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
Bases: {1,2,3,4,5,6,9,10,11} {1,2,3,4,5,6,8,10,11} {1,2,3,4,5,6,7,10,11} {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} {1,2,3,4,5,6,8,9,10} {1,2,3,4,5,6,7,9,10} {1,2,3,4,5,6,7,8,10} {1,2,3,4,5,6,7,8,9}
Loops: 1, 2, 3, 4, 5, 6. Parallel classes of size > 1: none.
Downloads
- Record as JSON
- OSCAR:
matroid_from_revlex_basis_encoding("000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000*000000**000000***000000****", 3, 12)