Matroid 3.11.48156
| Label | 3.11.48156 |
|---|---|
| Id | r3_n11_0******0******0***0******0****************0***********************0****************0***************0*******0*******0****************************0****0***********0*** |
| Rank | 3 |
| n | 11 |
Basic invariants
Representability
| Characteristic set | characteristic 0 and all primes except 3 [0,3] |
|---|---|
| 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 | 0 |
| Expected dimension over ℤ | 1 |
| Components of realization space | not computed |
| Free realization space | no |
| Principal ideal | yes |
| Birational type | point over Q(a)/(a^3 - a^2 + 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 - a^2 + 2*a + 1)",
"field": "Q(a)/(a^3 - a^2 + 2*a + 1)",
"minpoly": "T^3 - T^2 + 2*T + 1",
"degree": 3,
"core_vars": [
"x1"
],
"core_gens": [
"3*x1^3 - 16*x1^2 + 27*x1 - 15"
]
}
] |
| Good basis | 1,2,4 |
| Good basis v2 | 1,4,9 |
Geometry
| Realization space: scheme simple core id | r3_n11_0******0******0***0******0****************0***********************0****************0***************0*******0*******0****************************0****0***********0*** |
|---|---|
| Smooth char 0 | yes |
| Singular primes | [29] |
| Realization space: smooth over ℚ | not computed |
| Realization space: smooth over ℤ | no |
| Realization space: is regular scheme | yes how determined: regularity method: direct_mixed_jacobian |
| Realization space: singular fiber primes | [29] how determined: singular fiber primes method: terminal_three_lines_integer_certificate |
| Realization space: singular fiber characteristics | not computed |
| Realization space: characteristic dimensions | show[
{
"p": 0,
"d": 0
}
]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 | not computed |
Tutte polynomial
\(T = x_{0}^{3} + 8 x_{0}^{2} + 14 x_{0} x_{1} + 22 x_{0} + x_{1}^{8} + 3 x_{1}^{7} + 6 x_{1}^{6} + 10 x_{1}^{5} + 15 x_{1}^{4} + 21 x_{1}^{3} + 28 x_{1}^{2} + 22 x_{1}\)
Realization space
| Ring | \(\mathbb{Z}[x_{1}]\) |
|---|---|
| Defining ideal | \(\left(3 x_{1}^{3} - 16 x_{1}^{2} + 27 x_{1} - 15\right)\) |
| Inequations | \(3 x_{1}^{4} - 13 x_{1}^{3} + 20 x_{1}^{2} - 12 x_{1} + 3 \neq 0\), \(12 x_{1}^{4} - 58 x_{1}^{3} + 85 x_{1}^{2} - 39 x_{1} - 3 \neq 0\), \(6 x_{1}^{4} - 32 x_{1}^{3} + 54 x_{1}^{2} - 30 x_{1} + 3 \neq 0\), \(6 x_{1}^{3} - 23 x_{1}^{2} + 24 x_{1} - 3 \neq 0\), \(3 x_{1}^{3} - 7 x_{1}^{2} + 6 x_{1} - 3 \neq 0\), \(12 x_{1}^{3} - 46 x_{1}^{2} + 39 x_{1} + 3 \neq 0\), \(x_{1} \neq 0\), \(3 x_{1}^{2} - 7 x_{1} + 3 \neq 0\), \(6 x_{1}^{2} - 14 x_{1} + 3 \neq 0\), \(3 x_{1}^{2} - 16 x_{1} + 18 \neq 0\), \(36 x_{1}^{5} - 222 x_{1}^{4} + 457 x_{1}^{3} - 333 x_{1}^{2} + 24 x_{1} + 18 \neq 0\), \(12 x_{1}^{2} - 46 x_{1} + 39 \neq 0\), \(2 x_{1} - 3 \neq 0\), \(18 x_{1}^{4} - 84 x_{1}^{3} + 107 x_{1}^{2} - 30 x_{1} - 6 \neq 0\), \(3 x_{1}^{2} - 7 x_{1} + 6 \neq 0\), \(6 x_{1}^{3} - 29 x_{1}^{2} + 47 x_{1} - 27 \neq 0\), \(6 x_{1}^{3} - 23 x_{1}^{2} + 24 x_{1} - 6 \neq 0\), \(6 x_{1}^{4} - 32 x_{1}^{3} + 54 x_{1}^{2} - 33 x_{1} + 6 \neq 0\), \(18 x_{1}^{3} - 90 x_{1}^{2} + 139 x_{1} - 69 \neq 0\), \(12 x_{1}^{4} - 58 x_{1}^{3} + 85 x_{1}^{2} - 36 x_{1} - 6 \neq 0\), \(15 x_{1}^{3} - 71 x_{1}^{2} + 105 x_{1} - 51 \neq 0\), \(6 x_{1}^{2} - 23 x_{1} + 24 \neq 0\), \(3 \neq 0\), \(15 x_{1}^{2} - 53 x_{1} + 45 \neq 0\), \(18 x_{1}^{5} - 111 x_{1}^{4} + 260 x_{1}^{3} - 294 x_{1}^{2} + 174 x_{1} - 54 \neq 0\), \(12 x_{1}^{3} - 64 x_{1}^{2} + 108 x_{1} - 63 \neq 0\), \(18 x_{1}^{4} - 111 x_{1}^{3} + 245 x_{1}^{2} - 232 x_{1} + 81 \neq 0\), \(36 x_{1}^{5} - 222 x_{1}^{4} + 511 x_{1}^{3} - 603 x_{1}^{2} + 441 x_{1} - 189 \neq 0\), \(36 x_{1}^{5} - 222 x_{1}^{4} + 511 x_{1}^{3} - 594 x_{1}^{2} + 420 x_{1} - 171 \neq 0\) |
| Realization matrix | \(\begin{pmatrix}1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 1 \\ 0 & 1 & 3 x_{1}^{2} - 7 x_{1} + 5 & 1 & -6 x_{1}^{2} + 23 x_{1} - 20 & 0 & 3 x_{1}^{2} - 10 x_{1} + 10 & 3 x_{1}^{2} - 10 x_{1} + 10 & 0 & 1 & 1 \\ 0 & 1 & 3 x_{1}^{2} - 7 x_{1} + 5 & 0 & 0 & 1 & 3 x_{1}^{2} - 7 x_{1} + 5 & x_{1} & 1 & x_{1} - 1 & x_{1}\end{pmatrix}\) |
Combinatorics
Computed on the fly from the id.
Bases (151)
{1,2,4} {1,3,4} {2,3,4} {1,2,5} {1,3,5} {2,3,5} {2,4,5} {3,4,5} {1,2,6} {1,3,6} {2,3,6} {1,4,6} {3,4,6} {1,5,6} {2,5,6} {4,5,6} {1,2,7} {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} {1,2,8} {1,3,8} {2,3,8} {1,4,8} {2,4,8} {3,4,8} {1,5,8} {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} {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} {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} {2,7,10} {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} {1,9,10} {2,9,10} {3,9,10} {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} {5,8,11} {6,8,11} {7,8,11} {1,9,11} {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} {7,10,11} {8,10,11} {9,10,11}Non-bases (14)
{1,2,3} {1,4,5} {2,4,6} {3,5,6} {3,4,7} {2,5,8} {1,6,9} {7,8,9} {1,7,10} {3,8,10} {4,9,10} {4,8,11} {2,9,11} {6,10,11}Circuits (232)
{1,2,3} {1,4,5} {2,3,4,5} {2,4,6} {1,3,4,6} {1,2,5,6} {3,5,6} {1,2,4,7} {3,4,7} {1,2,5,7} {1,3,5,7} {2,3,5,7} {2,4,5,7} {1,2,6,7} {1,3,6,7} {2,3,6,7} {1,4,6,7} {1,5,6,7} {2,5,6,7} {4,5,6,7} {1,2,4,8} {1,3,4,8} {2,3,4,8} {2,5,8} {1,3,5,8} {3,4,5,8} {1,2,6,8} {1,3,6,8} {2,3,6,8} {1,4,6,8} {3,4,6,8} {1,5,6,8} {4,5,6,8} {1,2,7,8} {1,3,7,8} {2,3,7,8} {1,4,7,8} {2,4,7,8} {1,5,7,8} {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} {1,2,4,9} {1,3,4,9} {2,3,4,9} {1,2,5,9} {1,3,5,9} {2,3,5,9} {2,4,5,9} {3,4,5,9} {1,6,9} {2,3,6,9} {3,4,6,9} {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} {1,5,7,9} {2,5,7,9} {3,5,7,9} {4,5,7,9} {2,6,7,9} {3,6,7,9} {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} {3,5,8,9} {4,5,8,9} {2,6,8,9} {3,6,8,9} {4,6,8,9} {5,6,8,9} {7,8,9} {1,2,4,10} {1,3,4,10} {2,3,4,10} {1,2,5,10} {1,3,5,10} {2,3,5,10} {2,4,5,10} {3,4,5,10} {1,2,6,10} {1,3,6,10} {2,3,6,10} {1,4,6,10} {3,4,6,10} {1,5,6,10} {2,5,6,10} {4,5,6,10} {1,7,10} {2,3,7,10} {2,4,7,10} {2,5,7,10} {3,5,7,10} {4,5,7,10} {2,6,7,10} {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} {4,5,8,10} {1,6,8,10} {2,6,8,10} {4,6,8,10} {5,6,8,10} {2,7,8,10} {4,7,8,10} {5,7,8,10} {6,7,8,10} {1,2,9,10} {1,3,9,10} {2,3,9,10} {4,9,10} {1,5,9,10} {2,5,9,10} {3,5,9,10} {2,6,9,10} {3,6,9,10} {5,6,9,10} {2,7,9,10} {3,7,9,10} {5,7,9,10} {6,7,9,10} {1,8,9,10} {2,8,9,10} {5,8,9,10} {6,8,9,10} {1,2,4,11} {1,3,4,11} {2,3,4,11} {1,2,5,11} {1,3,5,11} {2,3,5,11} {2,4,5,11} {3,4,5,11} {1,2,6,11} {1,3,6,11} {2,3,6,11} {1,4,6,11} {3,4,6,11} {1,5,6,11} {2,5,6,11} {4,5,6,11} {1,2,7,11} {1,3,7,11} {2,3,7,11} {1,4,7,11} {2,4,7,11} {1,5,7,11} {2,5,7,11} {3,5,7,11} {4,5,7,11} {1,6,7,11} {2,6,7,11} {3,6,7,11} {4,6,7,11} {5,6,7,11} {1,2,8,11} {1,3,8,11} {2,3,8,11} {4,8,11} {1,5,8,11} {3,5,8,11} {1,6,8,11} {2,6,8,11} {3,6,8,11} {5,6,8,11} {1,7,8,11} {2,7,8,11} {3,7,8,11} {5,7,8,11} {6,7,8,11} {2,9,11} {1,3,9,11} {1,4,9,11} {3,4,9,11} {1,5,9,11} {3,5,9,11} {4,5,9,11} {3,6,9,11} {4,6,9,11} {5,6,9,11} {1,7,9,11} {3,7,9,11} {4,7,9,11} {5,7,9,11} {6,7,9,11} {1,8,9,11} {3,8,9,11} {5,8,9,11} {6,8,9,11} {1,2,10,11} {1,3,10,11} {2,3,10,11} {1,4,10,11} {2,4,10,11} {3,4,10,11} {1,5,10,11} {2,5,10,11} {3,5,10,11} {4,5,10,11} {6,10,11} {2,7,10,11} {3,7,10,11} {4,7,10,11} {5,7,10,11} {1,8,10,11} {2,8,10,11} {5,8,10,11} {7,8,10,11} {1,9,10,11} {3,9,10,11} {5,9,10,11} {7,9,10,11} {8,9,10,11}Flats by rank (40)
- rank 0 (1): {}
- rank 1 (11): {1} {2} {3} {4} {5} {6} {7} {8} {9} {10} {11}
- rank 2 (27): {1,2,3} {1,4,5} {2,4,6} {3,5,6} {2,7} {3,4,7} {5,7} {6,7} {1,8} {2,5,8} {6,8} {3,9} {5,9} {1,6,9} {7,8,9} {2,10} {5,10} {1,7,10} {3,8,10} {4,9,10} {1,11} {3,11} {5,11} {7,11} {4,8,11} {2,9,11} {6,10,11}
- rank 3 (1): {1,2,3,4,5,6,7,8,9,10,11}
Hyperplanes (27)
{1,2,3} {1,4,5} {2,4,6} {3,5,6} {2,7} {3,4,7} {5,7} {6,7} {1,8} {2,5,8} {6,8} {3,9} {5,9} {1,6,9} {7,8,9} {2,10} {5,10} {1,7,10} {3,8,10} {4,9,10} {1,11} {3,11} {5,11} {7,11} {4,8,11} {2,9,11} {6,10,11}Lines (14)
{1,2,3} {1,4,5} {2,4,6} {3,5,6} {3,4,7} {2,5,8} {1,6,9} {7,8,9} {1,7,10} {3,8,10} {4,9,10} {4,8,11} {2,9,11} {6,10,11}Dual (rank 8)
Revlex encoding in this labeling (not canonicalized, so not linked): ***0***********0****0****************************0*******0*******0***************0****************0***********************0****************0******0***0******0******0
Bases: {3,5,6,7,8,9,10,11} {2,5,6,7,8,9,10,11} {1,5,6,7,8,9,10,11} {3,4,6,7,8,9,10,11} {2,4,6,7,8,9,10,11} {1,4,6,7,8,9,10,11} {1,3,6,7,8,9,10,11} {1,2,6,7,8,9,10,11} {3,4,5,7,8,9,10,11} {2,4,5,7,8,9,10,11} {1,4,5,7,8,9,10,11} {2,3,5,7,8,9,10,11} {1,2,5,7,8,9,10,11} {2,3,4,7,8,9,10,11} {1,3,4,7,8,9,10,11} {1,2,3,7,8,9,10,11} {3,4,5,6,8,9,10,11} {2,4,5,6,8,9,10,11} {1,4,5,6,8,9,10,11} {2,3,5,6,8,9,10,11} {1,3,5,6,8,9,10,11} {2,3,4,6,8,9,10,11} {1,3,4,6,8,9,10,11} {1,2,4,6,8,9,10,11} {1,2,3,6,8,9,10,11} {2,3,4,5,8,9,10,11} {1,3,4,5,8,9,10,11} {1,2,4,5,8,9,10,11} {1,2,3,5,8,9,10,11} {1,2,3,4,8,9,10,11} {3,4,5,6,7,9,10,11} {2,4,5,6,7,9,10,11} {1,4,5,6,7,9,10,11} {2,3,5,6,7,9,10,11} {1,3,5,6,7,9,10,11} {1,2,5,6,7,9,10,11} {2,3,4,6,7,9,10,11} {1,2,4,6,7,9,10,11} {1,2,3,6,7,9,10,11} {2,3,4,5,7,9,10,11} {1,3,4,5,7,9,10,11} {1,2,4,5,7,9,10,11} {1,2,3,5,7,9,10,11} {1,2,3,4,7,9,10,11} {2,3,4,5,6,9,10,11} {1,3,4,5,6,9,10,11} {1,2,4,5,6,9,10,11} {1,2,3,5,6,9,10,11} {1,2,3,4,6,9,10,11} {1,2,3,4,5,9,10,11} {3,4,5,6,7,8,10,11} {2,4,5,6,7,8,10,11} {1,4,5,6,7,8,10,11} {2,3,5,6,7,8,10,11} {1,3,5,6,7,8,10,11} {1,2,5,6,7,8,10,11} {2,3,4,6,7,8,10,11} {1,3,4,6,7,8,10,11} {1,2,4,6,7,8,10,11} {1,2,3,6,7,8,10,11} {1,3,4,5,7,8,10,11} {1,2,4,5,7,8,10,11} {1,2,3,5,7,8,10,11} {1,2,3,4,7,8,10,11} {2,3,4,5,6,8,10,11} {1,3,4,5,6,8,10,11} {1,2,4,5,6,8,10,11} {1,2,3,5,6,8,10,11} {1,2,3,4,6,8,10,11} {1,2,3,4,5,8,10,11} {2,3,4,5,6,7,10,11} {1,3,4,5,6,7,10,11} {1,2,4,5,6,7,10,11} {1,2,3,5,6,7,10,11} {1,2,3,4,6,7,10,11} {1,2,3,4,5,7,10,11} {3,4,5,6,7,8,9,11} {2,4,5,6,7,8,9,11} {1,4,5,6,7,8,9,11} {2,3,5,6,7,8,9,11} {1,3,5,6,7,8,9,11} {1,2,5,6,7,8,9,11} {2,3,4,6,7,8,9,11} {1,3,4,6,7,8,9,11} {1,2,4,6,7,8,9,11} {1,2,3,6,7,8,9,11} {2,3,4,5,7,8,9,11} {1,3,4,5,7,8,9,11} {1,2,4,5,7,8,9,11} {1,2,3,5,7,8,9,11} {1,2,3,4,7,8,9,11} {1,3,4,5,6,8,9,11} {1,2,4,5,6,8,9,11} {1,2,3,5,6,8,9,11} {1,2,3,4,6,8,9,11} {1,2,3,4,5,8,9,11} {2,3,4,5,6,7,9,11} {1,3,4,5,6,7,9,11} {1,2,3,5,6,7,9,11} {1,2,3,4,6,7,9,11} {1,2,3,4,5,7,9,11} {1,2,3,4,5,6,9,11} {2,3,4,5,6,7,8,11} {1,3,4,5,6,7,8,11} {1,2,4,5,6,7,8,11} {1,2,3,4,6,7,8,11} {1,2,3,4,5,7,8,11} {1,2,3,4,5,6,8,11} {1,2,3,4,5,6,7,11} {3,4,5,6,7,8,9,10} {2,4,5,6,7,8,9,10} {1,4,5,6,7,8,9,10} {2,3,5,6,7,8,9,10} {1,3,5,6,7,8,9,10} {1,2,5,6,7,8,9,10} {2,3,4,6,7,8,9,10} {1,3,4,6,7,8,9,10} {1,2,4,6,7,8,9,10} {1,2,3,6,7,8,9,10} {2,3,4,5,7,8,9,10} {1,3,4,5,7,8,9,10} {1,2,4,5,7,8,9,10} {1,2,3,5,7,8,9,10} {1,2,3,4,7,8,9,10} {2,3,4,5,6,8,9,10} {1,3,4,5,6,8,9,10} {1,2,4,5,6,8,9,10} {1,2,3,5,6,8,9,10} {1,2,3,4,6,8,9,10} {1,2,3,4,5,8,9,10} {2,3,4,5,6,7,9,10} {1,3,4,5,6,7,9,10} {1,2,4,5,6,7,9,10} {1,2,3,4,6,7,9,10} {1,2,3,4,5,7,9,10} {1,2,3,4,5,6,9,10} {2,3,4,5,6,7,8,10} {1,2,4,5,6,7,8,10} {1,2,3,5,6,7,8,10} {1,2,3,4,6,7,8,10} {1,2,3,4,5,7,8,10} {1,2,3,4,5,6,8,10} {1,2,3,4,5,6,7,10} {2,3,4,5,6,7,8,9} {1,3,4,5,6,7,8,9} {1,2,4,5,6,7,8,9} {1,2,3,5,6,7,8,9} {1,2,3,4,6,7,8,9} {1,2,3,4,5,6,8,9} {1,2,3,4,5,6,7,9} {1,2,3,4,5,6,7,8}
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****0***********0***", 3, 11)