Matroid 3.11.9845
| Label | 3.11.9845 |
|---|---|
| Id | r3_n11_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 [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 | 2 |
| Expected dimension over ℤ | 3 |
| Components of realization space | not computed |
| Free realization space | no |
| Principal ideal | yes |
| Birational type | K3 surface how determined: birational type method: surface_elliptic_fibration_chi |
| Birational type components | show[
{
"dim": 2,
"free_rank": 0,
"torus_rank": 0,
"core_dim": 2,
"qbar_components": 1,
"type": "K3 surface",
"elliptic_fibration": {
"chi": 2,
"over": "x1",
"singular_fibres": [
"I6",
"I8",
"I1 (deg 3 place)",
"I1*"
],
"euler_number": 24,
"euler_check": true,
"shioda_tate_lower_bound_rho": 19,
"j_constant": false
},
"core_vars": [
"x1",
"x2",
"x3"
],
"core_gens": [
"x1^2*x2^2 - x1*x2^2*x3 - x1^2*x2 + x1*x3^2 + x2*x3 - x3^2"
]
}
] |
| Good basis | 2,4,7 |
| Good basis v2 | 2,4,8 |
Geometry
Other
| Char poly splits | false |
|---|---|
| Simplicial arrangement | no |
| Realization space: n qbar components | not computed |
Tutte polynomial
\(T = x_{0}^{3} + 8 x_{0}^{2} + 12 x_{0} x_{1} + 24 x_{0} + x_{1}^{8} + 3 x_{1}^{7} + 6 x_{1}^{6} + 10 x_{1}^{5} + 15 x_{1}^{4} + 21 x_{1}^{3} + 28 x_{1}^{2} + 24 x_{1}\)
Realization space
| Ring | \(\mathbb{Z}[x_{1}, x_{2}, x_{3}]\) |
|---|---|
| Defining ideal | \(\left(x_{1}^{2} x_{2}^{2} - x_{1}^{2} x_{2} - x_{1} x_{2}^{2} x_{3} + x_{1} x_{3}^{2} + x_{2} x_{3} - x_{3}^{2}\right)\) |
| Inequations | \(x_{1} - x_{3} \neq 0\), \(x_{2} - x_{3} \neq 0\), \(x_{2} \neq 0\), \(x_{1} x_{2}^{2} - x_{2}^{2} x_{3} - x_{2} x_{3} + x_{3}^{2} \neq 0\), \(x_{3} - 1 \neq 0\), \(x_{3} \neq 0\), \(x_{2} - 1 \neq 0\), \(x_{1} x_{2} - x_{2} x_{3} - x_{2} + x_{3} \neq 0\), \(x_{1} x_{2}^{2} - 2 x_{1} x_{2} + x_{1} x_{3} - x_{2}^{2} x_{3} + x_{2} x_{3} + x_{2} - x_{3} \neq 0\), \(x_{1} x_{2}^{2} - x_{1} x_{2} - x_{2}^{2} x_{3} + x_{2} + x_{3}^{2} - x_{3} \neq 0\), \(x_{1} - 1 \neq 0\), \(x_{1} x_{2} - x_{2} - x_{3}^{2} + x_{3} \neq 0\), \(x_{1} x_{2} - x_{3} \neq 0\), \(x_{1} \neq 0\), \(x_{1}^{2} x_{2} - x_{1} x_{2} x_{3} - x_{2} x_{3} + x_{3}^{2} \neq 0\), \(x_{1}^{2} x_{2}^{2} - x_{1}^{2} x_{2} + x_{1}^{2} x_{3} - x_{1} x_{2}^{2} x_{3} - x_{1} x_{2} x_{3} + x_{2} x_{3}^{2} + x_{2} x_{3} - x_{3}^{2} \neq 0\), \(x_{1} - x_{2} \neq 0\), \(x_{1}^{2} - x_{3} \neq 0\), \(x_{1}^{2} x_{2} - x_{1} x_{2} x_{3} + x_{1} x_{3} - x_{2} x_{3} \neq 0\), \(x_{1}^{2} x_{2} - x_{1} x_{3}^{2} - x_{2} x_{3} + x_{3}^{2} \neq 0\), \(x_{1} x_{2} - x_{1} x_{3} - x_{1} + x_{3} \neq 0\), \(x_{1}^{2} x_{2}^{2} - x_{1} x_{2}^{2} x_{3} - x_{2} x_{3}^{2} + x_{3}^{3} \neq 0\), \(x_{1}^{2} x_{2}^{2} - x_{1}^{2} x_{2} - x_{1} x_{2}^{2} x_{3} + x_{1} x_{2} x_{3} - x_{2} x_{3}^{2} + x_{2} x_{3} + x_{3}^{3} - x_{3}^{2} \neq 0\), \(x_{1}^{2} x_{2} - x_{1} x_{2} x_{3} - x_{1} x_{3}^{2} + x_{2} x_{3}^{2} - x_{2} x_{3} + x_{3}^{2} \neq 0\), \(x_{1}^{2} x_{2}^{2} - x_{1} x_{2}^{2} x_{3} - x_{1} x_{2} x_{3}^{2} - x_{2}^{2} x_{3} + x_{2} x_{3}^{3} + 2 x_{2} x_{3}^{2} - x_{3}^{3} \neq 0\), \(x_{1}^{2} x_{2} - x_{1} x_{2} x_{3} + x_{2} x_{3}^{2} - x_{2} x_{3} - x_{3}^{3} + x_{3}^{2} \neq 0\), \(x_{1} x_{2} - x_{1} - x_{3}^{2} + x_{3} \neq 0\), \(x_{1} x_{2} - x_{3}^{2} \neq 0\), \(x_{1} x_{2} + x_{2}^{2} - 2 x_{2} x_{3} - x_{2} + x_{3} \neq 0\) |
| Realization matrix | \(\begin{pmatrix}1 & 1 & 1 & 0 & 1 & x_{1} - x_{3} & x_{1} - x_{3} & 0 & x_{3} & x_{3} & 1 \\ 1 & 0 & x_{2} & 1 & 0 & x_{2} - x_{3} & x_{2} - x_{3} & 0 & x_{1} x_{2} & x_{1} x_{2} & x_{2} \\ 1 & 0 & x_{2} & 0 & 1 & 0 & x_{1} x_{2} - x_{2} x_{3} & 1 & x_{3}^{2} & x_{1} x_{3} & x_{3}\end{pmatrix}\) |
Combinatorics
Computed on the fly from the id.
Bases (153)
{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} {3,5,6} {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} {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} {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} {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} {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} {4,8,11} {5,8,11} {6,8,11} {7,8,11} {1,9,11} {2,9,11} {3,9,11} {5,9,11} {6,9,11} {7,9,11} {8,9,11} {1,10,11} {3,10,11} {4,10,11} {5,10,11} {6,10,11} {7,10,11} {8,10,11} {9,10,11}Non-bases (12)
{1,2,3} {1,4,5} {2,4,6} {3,4,7} {2,5,8} {6,7,8} {1,7,9} {5,7,10} {8,9,10} {3,8,11} {4,9,11} {2,10,11}Circuits (246)
{1,2,3} {1,4,5} {2,3,4,5} {2,4,6} {1,3,4,6} {1,2,5,6} {1,3,5,6} {2,3,5,6} {3,4,5,6} {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} {3,5,6,7} {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} {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} {1,5,7,8} {3,5,7,8} {4,5,7,8} {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} {1,2,6,9} {1,3,6,9} {2,3,6,9} {1,4,6,9} {3,4,6,9} {1,5,6,9} {2,5,6,9} {3,5,6,9} {4,5,6,9} {1,7,9} {2,3,7,9} {2,4,7,9} {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} {1,6,8,9} {2,6,8,9} {3,6,8,9} {4,6,8,9} {5,6,8,9} {2,7,8,9} {3,7,8,9} {4,7,8,9} {5,7,8,9} {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} {3,5,6,10} {4,5,6,10} {1,2,7,10} {1,3,7,10} {2,3,7,10} {1,4,7,10} {2,4,7,10} {5,7,10} {1,6,7,10} {2,6,7,10} {3,6,7,10} {4,6,7,10} {1,2,8,10} {1,3,8,10} {2,3,8,10} {1,4,8,10} {2,4,8,10} {3,4,8,10} {1,5,8,10} {3,5,8,10} {4,5,8,10} {1,6,8,10} {2,6,8,10} {3,6,8,10} {4,6,8,10} {5,6,8,10} {1,7,8,10} {2,7,8,10} {3,7,8,10} {4,7,8,10} {1,2,9,10} {1,3,9,10} {2,3,9,10} {1,4,9,10} {2,4,9,10} {3,4,9,10} {1,5,9,10} {2,5,9,10} {3,5,9,10} {4,5,9,10} {1,6,9,10} {2,6,9,10} {3,6,9,10} {4,6,9,10} {5,6,9,10} {2,7,9,10} {3,7,9,10} {4,7,9,10} {6,7,9,10} {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} {3,5,6,11} {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} {3,8,11} {1,4,8,11} {2,4,8,11} {1,5,8,11} {4,5,8,11} {1,6,8,11} {2,6,8,11} {4,6,8,11} {5,6,8,11} {1,7,8,11} {2,7,8,11} {4,7,8,11} {5,7,8,11} {1,2,9,11} {1,3,9,11} {2,3,9,11} {4,9,11} {1,5,9,11} {2,5,9,11} {3,5,9,11} {1,6,9,11} {2,6,9,11} {3,6,9,11} {5,6,9,11} {2,7,9,11} {3,7,9,11} {5,7,9,11} {6,7,9,11} {1,8,9,11} {2,8,9,11} {5,8,9,11} {6,8,9,11} {7,8,9,11} {2,10,11} {1,3,10,11} {1,4,10,11} {3,4,10,11} {1,5,10,11} {3,5,10,11} {4,5,10,11} {1,6,10,11} {3,6,10,11} {4,6,10,11} {5,6,10,11} {1,7,10,11} {3,7,10,11} {4,7,10,11} {6,7,10,11} {1,8,10,11} {4,8,10,11} {5,8,10,11} {6,8,10,11} {7,8,10,11} {1,9,10,11} {3,9,10,11} {5,9,10,11} {6,9,10,11} {7,9,10,11}Flats by rank (44)
- rank 0 (1): {}
- rank 1 (11): {1} {2} {3} {4} {5} {6} {7} {8} {9} {10} {11}
- rank 2 (31): {1,2,3} {3,5} {1,4,5} {1,6} {3,6} {2,4,6} {5,6} {2,7} {3,4,7} {1,8} {4,8} {2,5,8} {6,7,8} {2,9} {3,9} {5,9} {6,9} {1,7,9} {1,10} {3,10} {4,10} {6,10} {5,7,10} {8,9,10} {1,11} {5,11} {6,11} {7,11} {3,8,11} {4,9,11} {2,10,11}
- rank 3 (1): {1,2,3,4,5,6,7,8,9,10,11}
Hyperplanes (31)
{1,2,3} {3,5} {1,4,5} {1,6} {3,6} {2,4,6} {5,6} {2,7} {3,4,7} {1,8} {4,8} {2,5,8} {6,7,8} {2,9} {3,9} {5,9} {6,9} {1,7,9} {1,10} {3,10} {4,10} {6,10} {5,7,10} {8,9,10} {1,11} {5,11} {6,11} {7,11} {3,8,11} {4,9,11} {2,10,11}Lines (12)
{1,2,3} {1,4,5} {2,4,6} {3,4,7} {2,5,8} {6,7,8} {1,7,9} {5,7,10} {8,9,10} {3,8,11} {4,9,11} {2,10,11}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
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} {1,2,4,7,8,9,10,11} {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} {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} {2,3,4,5,7,8,10,11} {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} {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} {1,2,3,4,5,6,10,11} {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} {2,3,4,5,6,8,9,11} {1,3,4,5,6,8,9,11} {1,2,4,5,6,8,9,11} {1,2,3,5,6,8,9,11} {1,2,3,4,5,8,9,11} {2,3,4,5,6,7,9,11} {1,3,4,5,6,7,9,11} {1,2,4,5,6,7,9,11} {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} {1,2,3,5,6,7,8,11} {1,2,3,4,6,7,8,11} {1,2,3,4,5,7,8,11} {1,2,3,4,5,6,8,11} {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} {1,2,3,5,6,7,9,10} {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} {1,3,4,5,6,7,8,10} {1,2,4,5,6,7,8,10} {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} {1,2,4,5,6,7,8,9} {1,2,3,5,6,7,8,9} {1,2,3,4,6,7,8,9} {1,2,3,4,5,7,8,9} {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*******", 3, 11)