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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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1,370,992 matroids, showing 19,351–19,400

label ▲	rank ⇅	n ⇅	#bases ⇅	#automorphisms ⇅	simple ⇅	char set ⇅	rs dim ⇅	birational type ⇅
3.11.262894	3	11	141	6	✓	all	5	rational
3.11.262902	3	11	139	2	✓	all	5	rational
3.11.262915	3	11	140	1	✓	all	5	rational
3.11.262932	3	11	139	2	✓	all	5	rational
3.11.262957	3	11	137	16	✓	all	5	point over Q(a)/(a^2 + 1)
3.11.262979	3	11	144	96	✗	all	5	rational
3.11.263003	3	11	142	24	✗	all	5	rational
3.11.263004	3	11	142	24	✗	all	5	rational
3.11.263015	3	11	138	96	✗	all	5	rational
3.11.263037	3	11	138	8	✗	all	5	rational
3.11.263041	3	11	135	960	✗	all	5	rational
3.11.263060	3	11	142	48	✗	all	5	rational
3.11.263068	3	11	135	96	✗	all	5	rational
3.11.263100	3	11	140	4	✗	all	5	rational
3.11.263105	3	11	141	12	✗	all	5	rational
3.11.263113	3	11	141	4	✗	all	5	rational
3.11.263125	3	11	140	4	✗	all	5	rational
3.11.263130	3	11	141	4	✗	all	5	rational
3.11.263138	3	11	138	4	✗	all	5	rational
3.11.263143	3	11	138	2	✗	all	5	rational
3.11.263151	3	11	140	2	✗	all	5	rational
3.11.263155	3	11	140	2	✗	all	5	rational
3.11.263159	3	11	140	4	✗	all	5	rational
3.11.263168	3	11	138	4	✗	all	5	rational
3.11.263183	3	11	137	2	✗	all	5	rational
3.11.263194	3	11	139	12	✗	all	5	rational
3.11.263198	3	11	137	4	✗	all	5	rational
3.11.263212	3	11	141	24	✗	all	5	rational
3.11.263226	3	11	140	4	✗	all	5	rational
3.11.263233	3	11	139	4	✗	all	5	rational
3.11.263238	3	11	139	4	✗	all	5	rational
3.11.263239	3	11	140	4	✗	all	5	rational
3.11.263249	3	11	139	4	✗	all	5	rational
3.11.263256	3	11	137	2	✗	all	5	rational
3.11.263260	3	11	137	2	✗	all	5	rational
3.11.263263	3	11	138	2	✗	all	5	rational
3.11.263265	3	11	136	4	✗	all	5	rational
3.11.263267	3	11	138	8	✗	all	5	rational
3.11.263272	3	11	136	4	✗	all	5	rational
3.11.263283	3	11	140	4	✗	all	5	rational
3.11.263291	3	11	139	4	✗	all	5	rational
3.11.263292	3	11	138	4	✗	all	5	rational
3.11.263299	3	11	138	24	✗	all	5	rational
3.11.263302	3	11	138	4	✗	all	5	rational
3.11.263305	3	11	139	4	✗	all	5	rational
3.11.263331	3	11	138	4	✗	all	5	rational
3.11.263346	3	11	137	24	✗	all	5	rational
3.11.263369	3	11	137	12	✗	all	5	rational
3.11.263386	3	11	137	12	✗	all	5	rational
3.11.263392	3	11	140	16	✗	all	5	rational