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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
Properties

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8,076,768 matroids, showing 20,951–21,000

label ▲rank ⇅n ⇅#bases ⇅#automorphisms ⇅simple ⇅char set ⇅rs dim ⇅birational type ⇅
3.11.929613111501✓all2genus 1 curve (11a3)
3.11.929623111521✓all2rational
3.11.929633111511✓all2rational
3.11.929643111511✓all but 22rational
3.11.929673111511✓all2rational
3.11.929803111521✓all but 22rational
3.11.929853111511✓all2rational
3.11.929863111501✓all2rational
3.11.930293111501✓all2rational
3.11.930303111501✓all2rational
3.11.930313111531✓all2rational
3.11.930323111521✓all2rational
3.11.930343111521✓all2rational
3.11.930353111521✓all2rational
3.11.930363111521✓all2rational
3.11.930373111521✓all2rational
3.11.930453111511✓all but 22rational
3.11.930463111521✓all but 22rational
3.11.930473111511✓all but 22rational
3.11.930493111511✓all2rational
3.11.930503111511✓all2rational
3.11.930513111521✓all2rational
3.11.930523111511✓all2rational
3.11.930543111511✓all2rational
3.11.930553111521✓all2rational
3.11.930563111511✓all2rational
3.11.930643111521✓all but 22rational
3.11.930653111511✓all but 22rational
3.11.930663111511✓all but 22rational
3.11.930683111511✓all2rational
3.11.930693111501✓all2rational
3.11.930703111511✓all2rational
3.11.930713111501✓all2rational
3.11.930733111511✓all2rational
3.11.930743111501✓all2rational
3.11.930753111501✓all2empty
3.11.930953111501✓all2rational
3.11.930963111511✓all2rational
3.11.930973111501✓all2rational
3.11.930983111511✓all2rational
3.11.931003111501✓all2rational
3.11.931013111501✓all2rational
3.11.931023111491✓all2point over Q(a)/(a^2 - 2)
3.11.931033111491✓all2rational
3.11.931043111501✓all2empty
3.11.931063111521✓all2rational
3.11.931073111521✓all2rational
3.11.931083111521✓all2rational
3.11.931093111521✓all2rational
3.11.931113111521✓all2rational

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