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…or describe it by its bases, non-bases or lines

Any labeling of the ground set $\{1,\dots,n\}$ will do: we compute the canonical id. Sets are written as {1,2,3}, one per line, or, when $n \le 9$, as digit strings such as 123 145. Lines are the lines with at least three points of a simple rank-3 matroid. A revlex encoding is a string of * and 0.

Size rank n

Invariants Bases Automorphisms Dimension of realization space Girth Expected dimension over ℤ Components of realization space over ℚ Components of realization space over ℚ̄

Realizability realizable in char 0 realizable in characteristic char set class birational type

Properties Simple Connected Paving Uniform Regular Binary Ternary Quaternary Graphic Transversal Series parallel Self dual Orientable Supersolvable Three lines Free realization space Realization space: smooth over ℤ Realization space: is regular scheme

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32,400,859 matroids, showing 21,351–21,400

label ▼	rank ⇅	n ⇅	#bases ⇅	#automorphisms ⇅	simple ⇅	char set ⇅	rs dim ⇅	birational type ⇅
4.9.168863	4	9	110	2	✓	all but 2	2	rational
4.9.168862	4	9	109	1	✓	∅	-1	rational
4.9.168861	4	9	110	2	✓	all but 2	2	rational
4.9.168860	4	9	108	1	✓	all but 2,3	0	rational
4.9.168859	4	9	109	1	✓	all	3	rational
4.9.168858	4	9	109	1	✓	all	3	rational
4.9.168857	4	9	110	2	✓	all	1	rational
4.9.168856	4	9	109	1	✓	all	2	rational
4.9.168855	4	9	109	1	✓	all	4	rational
4.9.168854	4	9	110	1	✓	all but 2,5	0	rational
4.9.168853	4	9	109	1	✓	∅	-1	rational
4.9.168852	4	9	110	1	✓	all	1	rational
4.9.168851	4	9	110	1	✓	all	1	rational
4.9.168850	4	9	111	2	✓	all	1	rational
4.9.168849	4	9	106	8	✓	∅	-1	empty
4.9.168848	4	9	107	8	✓	all	3	empty
4.9.168847	4	9	106	2	✓	all	1	empty
4.9.168846	4	9	107	2	✓	all	2	empty
4.9.168845	4	9	107	2	✓	all but 2	2	empty
4.9.168844	4	9	108	2	✓	all but 2	1	empty
4.9.168843	4	9	107	2	✓	all but 2	1	empty
4.9.168842	4	9	108	2	✓	all	2	empty
4.9.168841	4	9	107	2	✓	all	3	point over Q(a)/(a^2 - 2)
4.9.168840	4	9	108	2	✓	∅	-1	empty
4.9.168839	4	9	107	2	✓	all	3	rational
4.9.168838	4	9	108	2	✓	all	3	empty
4.9.168837	4	9	108	2	✓	all	1	rational
4.9.168836	4	9	109	2	✓	all but 2	2	empty
4.9.168835	4	9	107	1	✓	all but 2	3	empty
4.9.168834	4	9	108	1	✓	all but 2	2	empty
4.9.168833	4	9	107	1	✓	∅	-1	empty
4.9.168832	4	9	107	1	✓	all but 2	1	empty
4.9.168831	4	9	108	1	✓	all but 2	2	rational
4.9.168830	4	9	107	1	✓	all	3	empty
4.9.168829	4	9	107	1	✓	{2}	-1	empty
4.9.168828	4	9	108	1	✓	all but 2	2	rational
4.9.168827	4	9	108	1	✓	all	3	point over Q(a)/(a^2 - a - 1)
4.9.168826	4	9	108	1	✓	all	2	empty
4.9.168825	4	9	109	1	✓	all	2	rational
4.9.168824	4	9	108	2	✓	all but 2	2	empty
4.9.168823	4	9	108	2	✓	all	1	empty
4.9.168822	4	9	109	2	✓	all but 2	3	empty
4.9.168821	4	9	108	1	✓	all	2	rational
4.9.168820	4	9	108	1	✓	∅	-1	empty
4.9.168819	4	9	109	1	✓	{2}	-1	rational
4.9.168818	4	9	108	1	✓	all	6	point over Q(a)/(a^2 - a - 4)
4.9.168817	4	9	108	1	✓	all but 2	2	empty
4.9.168816	4	9	109	1	✓	all	2	rational
4.9.168815	4	9	108	1	✓	all	7	rational
4.9.168814	4	9	108	1	✓	{2}	-1	empty