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

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.

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Realizability
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8,076,768 matroids, showing 20,501–20,550

label ▲rank ⇅n ⇅#bases ⇅#automorphisms ⇅simple ⇅char set ⇅rs dim ⇅birational type ⇅
3.11.916083111511✓all2rational
3.11.916093111511✓all2rational
3.11.916133111511✓all2rational
3.11.916143111511✓all2rational
3.11.916153111511✓all2rational
3.11.916163111511✓all2rational
3.11.916253111511✓all2rational
3.11.916263111511✓all2rational
3.11.916273111511✓all2rational
3.11.916333111511✓all2empty
3.11.916343111501✓all2empty
3.11.916353111531✓all2rational
3.11.916433111511✓all2rational
3.11.916483111511✓all2rational
3.11.916503111521✓all2rational
3.11.916523111511✓all2rational
3.11.916543111511✓all2rational
3.11.916553111521✓all2rational
3.11.916633111511✓all but 22empty
3.11.916643111501✓all2empty
3.11.916653111511✓all2rational
3.11.916663111501✓all2rational
3.11.916683111511✓all2rational
3.11.916693111501✓all2rational
3.11.916703111501✓all2rational
3.11.916713111521✓all2rational
3.11.916733111511✓all2rational
3.11.916743111511✓all2rational
3.11.916753111511✓all2rational
3.11.916763111501✓all2rational
3.11.916783111501✓all2rational
3.11.916793111501✓all2genus 1 curve (19a3)
3.11.916803111501✓all2rational
3.11.916813111492✓all2rational
3.11.916823111491✓all2rational
3.11.916863111511✓all2rational
3.11.916913111501✓all but 22rational
3.11.916923111501✓all2curve of genus 2
3.11.916933111511✓all2rational
3.11.916943111501✓all2genus 1 curve (11a3)
3.11.916993111501✓all2empty
3.11.917033111501✓all2rational
3.11.917083111511✓all2rational
3.11.917093111501✓all2rational
3.11.917103111501✓all2rational
3.11.917113111511✓all2empty
3.11.917123111501✓all2empty
3.11.917163111501✓all2rational
3.11.917213111511✓all2rational
3.11.917223111501✓all2rational

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