Matroid 3.10.6712
| Label | 3.10.6712 |
|---|---|
| Id | r3_n10_0000000000********************0***************0*********************0**************0****************0******0****0******* |
| Rank | 3 |
| n | 10 |
Basic invariants
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 | 3 |
| Components of realization space | 2 |
| Free realization space | no |
| Principal ideal | yes |
| Birational type | rational over Q(a)/(a^2 - a + 1) how determined: birational type method: point_field |
| Birational type components | show[
{
"dim": 2,
"free_rank": 2,
"torus_rank": 2,
"core_dim": 0,
"qbar_components": 2,
"type": "rational over Q(a)/(a^2 - a + 1)",
"field": "Q(a)/(a^2 - a + 1)",
"minpoly": "T^2 - T + 1",
"degree": 2,
"reason": "A^2 over the field of definition",
"core_vars": [
"x3"
],
"core_gens": [
"x3^2 - x3 + 1"
]
}
] |
| Good basis | 1,2,6 |
| Good basis v2 | 1,2,8 |
Geometry
| Realization space: scheme simple core id | r3_n10_0000000000********************0***************0*********************0**************0****************0******0****0******* |
|---|---|
| Smooth char 0 | yes |
| Singular primes | [3] |
| Realization space: smooth over ℚ | not computed |
| Realization space: smooth over ℤ | no |
| Realization space: is regular scheme | yes how determined: regularity method: three_lines_deletion regularity witness: 4=>r3_n9_0000************0**********0***************0***********0***********0*****0***0****** |
| Realization space: singular fiber primes | [3] how determined: singular fiber primes method: smooth_deletion_sandwich |
| Realization space: singular fiber characteristics | not computed |
| Realization space: characteristic dimensions | show[
{
"p": 0,
"d": 2
}
]how determined: characteristic dimensions method: cofinite_strong_groebner_candidates_exact |
| Realization space: characteristic dimension unexpected | no |
| Realization space: characteristic dimension varies | no |
Other
| Char poly splits | false |
|---|---|
| Simplicial arrangement | no |
| Realization space: n qbar components | 2 |
Tutte polynomial
\(T = x_{0}^{3} + 7 x_{0}^{2} + x_{0} x_{1}^{3} + 2 x_{0} x_{1}^{2} + 10 x_{0} x_{1} + 15 x_{0} + x_{1}^{7} + 3 x_{1}^{6} + 6 x_{1}^{5} + 10 x_{1}^{4} + 14 x_{1}^{3} + 18 x_{1}^{2} + 15 x_{1}\)
Realization space
| Ring | \(\mathbb{Z}[x_{1}, x_{2}, x_{3}]\) |
|---|---|
| Defining ideal | \(\left(x_{3}^{2} - x_{3} + 1\right)\) |
| Inequations | \(x_{1} - x_{3}^{2} + x_{3} - 1 \neq 0\), \(x_{2} - x_{3}^{2} + x_{3} - 1 \neq 0\), \(x_{1} - 1 \neq 0\), \(x_{2} - 1 \neq 0\), \(x_{1} + x_{3} - 1 \neq 0\), \(x_{2} + x_{3} - 1 \neq 0\), \(x_{2} \neq 0\), \(x_{1} - x_{2} \neq 0\), \(x_{1} \neq 0\), \(x_{1} - x_{3} \neq 0\), \(x_{2} - x_{3} \neq 0\) |
| Realization matrix | \(\begin{pmatrix}1 & 0 & 1 & 1 & 1 & 0 & 1 & 0 & 1 & 1 \\ 0 & 1 & 1 & x_{1} & x_{2} & 1 & x_{3} & 0 & x_{3} & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 & x_{3} & 1 & x_{3} - 1 & x_{3}\end{pmatrix}\) |
Combinatorics
Computed on the fly from the id.
Bases (103)
{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} {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} {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} {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} {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} {3,7,10} {4,7,10} {5,7,10} {6,7,10} {1,8,10} {2,8,10} {4,8,10} {5,8,10} {6,8,10} {7,8,10} {2,9,10} {3,9,10} {4,9,10} {5,9,10} {6,9,10} {7,9,10} {8,9,10}Non-bases (17)
{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} {1,6,7} {2,6,8} {3,6,9} {7,8,9} {2,7,10} {3,8,10} {1,9,10}Circuits (123)
{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} {1,6,7} {2,3,6,7} {2,4,6,7} {3,4,6,7} {2,5,6,7} {3,5,6,7} {4,5,6,7} {2,6,8} {1,3,6,8} {1,4,6,8} {3,4,6,8} {1,5,6,8} {3,5,6,8} {4,5,6,8} {1,2,7,8} {1,3,7,8} {2,3,7,8} {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} {3,6,7,8} {4,6,7,8} {5,6,7,8} {1,2,6,9} {3,6,9} {1,4,6,9} {2,4,6,9} {1,5,6,9} {2,5,6,9} {4,5,6,9} {1,2,7,9} {1,3,7,9} {2,3,7,9} {1,4,7,9} {2,4,7,9} {3,4,7,9} {1,5,7,9} {2,5,7,9} {3,5,7,9} {4,5,7,9} {2,6,7,9} {4,6,7,9} {5,6,7,9} {1,2,8,9} {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} {4,6,8,9} {5,6,8,9} {7,8,9} {1,2,6,10} {1,3,6,10} {2,3,6,10} {1,4,6,10} {2,4,6,10} {3,4,6,10} {1,5,6,10} {2,5,6,10} {3,5,6,10} {4,5,6,10} {2,7,10} {1,3,7,10} {1,4,7,10} {3,4,7,10} {1,5,7,10} {3,5,7,10} {4,5,7,10} {3,6,7,10} {4,6,7,10} {5,6,7,10} {1,2,8,10} {3,8,10} {1,4,8,10} {2,4,8,10} {1,5,8,10} {2,5,8,10} {4,5,8,10} {1,6,8,10} {4,6,8,10} {5,6,8,10} {1,7,8,10} {4,7,8,10} {5,7,8,10} {6,7,8,10} {1,9,10} {2,3,9,10} {2,4,9,10} {3,4,9,10} {2,5,9,10} {3,5,9,10} {4,5,9,10} {2,6,9,10} {4,6,9,10} {5,6,9,10} {3,7,9,10} {4,7,9,10} {5,7,9,10} {6,7,9,10} {2,8,9,10} {4,8,9,10} {5,8,9,10} {6,8,9,10}Flats by rank (34)
- rank 0 (1): {}
- rank 1 (10): {1} {2} {3} {4} {5} {6} {7} {8} {9} {10}
- rank 2 (22): {1,2,3,4,5} {4,6} {5,6} {3,7} {4,7} {5,7} {1,6,7} {1,8} {4,8} {5,8} {2,6,8} {2,9} {4,9} {5,9} {3,6,9} {7,8,9} {4,10} {5,10} {6,10} {2,7,10} {3,8,10} {1,9,10}
- rank 3 (1): {1,2,3,4,5,6,7,8,9,10}
Hyperplanes (22)
{1,2,3,4,5} {4,6} {5,6} {3,7} {4,7} {5,7} {1,6,7} {1,8} {4,8} {5,8} {2,6,8} {2,9} {4,9} {5,9} {3,6,9} {7,8,9} {4,10} {5,10} {6,10} {2,7,10} {3,8,10} {1,9,10}Lines (8)
{1,2,3,4,5} {1,6,7} {2,6,8} {3,6,9} {7,8,9} {2,7,10} {3,8,10} {1,9,10}Dual (rank 7)
Revlex encoding in this labeling (not canonicalized, so not linked): *******0****0******0****************0**************0*********************0***************0********************0000000000
Bases: {3,4,5,7,8,9,10} {2,4,5,7,8,9,10} {1,4,5,7,8,9,10} {2,3,5,7,8,9,10} {1,3,5,7,8,9,10} {1,2,5,7,8,9,10} {2,3,4,7,8,9,10} {1,3,4,7,8,9,10} {1,2,4,7,8,9,10} {1,2,3,7,8,9,10} {3,4,5,6,8,9,10} {2,4,5,6,8,9,10} {1,4,5,6,8,9,10} {2,3,5,6,8,9,10} {1,3,5,6,8,9,10} {1,2,5,6,8,9,10} {2,3,4,6,8,9,10} {1,3,4,6,8,9,10} {1,2,4,6,8,9,10} {1,2,3,6,8,9,10} {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} {3,4,5,6,7,9,10} {2,4,5,6,7,9,10} {1,4,5,6,7,9,10} {2,3,5,6,7,9,10} {1,3,5,6,7,9,10} {1,2,5,6,7,9,10} {2,3,4,6,7,9,10} {1,3,4,6,7,9,10} {1,2,4,6,7,9,10} {1,2,3,6,7,9,10} {2,3,4,5,7,9,10} {1,2,4,5,7,9,10} {1,2,3,5,7,9,10} {1,2,3,4,7,9,10} {2,3,4,5,6,9,10} {1,3,4,5,6,9,10} {1,2,4,5,6,9,10} {1,2,3,5,6,9,10} {1,2,3,4,6,9,10} {1,2,3,4,5,9,10} {3,4,5,6,7,8,10} {2,4,5,6,7,8,10} {1,4,5,6,7,8,10} {2,3,5,6,7,8,10} {1,3,5,6,7,8,10} {1,2,5,6,7,8,10} {2,3,4,6,7,8,10} {1,3,4,6,7,8,10} {1,2,4,6,7,8,10} {1,2,3,6,7,8,10} {2,3,4,5,7,8,10} {1,3,4,5,7,8,10} {1,2,3,5,7,8,10} {1,2,3,4,7,8,10} {2,3,4,5,6,8,10} {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} {1,2,3,4,5,8,10} {2,3,4,5,6,7,10} {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} {1,2,3,4,5,7,10} {3,4,5,6,7,8,9} {2,4,5,6,7,8,9} {1,4,5,6,7,8,9} {2,3,5,6,7,8,9} {1,3,5,6,7,8,9} {1,2,5,6,7,8,9} {2,3,4,6,7,8,9} {1,3,4,6,7,8,9} {1,2,4,6,7,8,9} {1,2,3,6,7,8,9} {2,3,4,5,7,8,9} {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} {2,3,4,5,6,8,9} {1,2,4,5,6,8,9} {1,2,3,5,6,8,9} {1,2,3,4,6,8,9} {1,2,3,4,5,8,9} {2,3,4,5,6,7,9} {1,3,4,5,6,7,9} {1,2,3,5,6,7,9} {1,2,3,4,6,7,9} {1,2,3,4,5,7,9} {1,2,3,4,5,6,9} {1,3,4,5,6,7,8} {1,2,4,5,6,7,8} {1,2,3,5,6,7,8} {1,2,3,4,6,7,8} {1,2,3,4,5,7,8} {1,2,3,4,5,6,8} {1,2,3,4,5,6,7}
Loops: none. Parallel classes of size > 1: none.
Downloads
- Record as JSON
- OSCAR:
matroid_from_revlex_basis_encoding("0000000000********************0***************0*********************0**************0****************0******0****0*******", 3, 10)