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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 20,901–20,950

label ▲rank ⇅n ⇅#bases ⇅#automorphisms ⇅simple ⇅char set ⇅rs dim ⇅birational type ⇅
3.11.928733111511✓all but 22rational
3.11.928743111531✓all but 22rational
3.11.928753111521✓all but 22rational
3.11.928763111521✓all but 22rational
3.11.928773111521✓all but 22rational
3.11.928803111521✓all but 22rational
3.11.928813111511✓all but 22rational
3.11.928823111511✓all but 22rational
3.11.928833111511✓all but 22rational
3.11.928853111511✓all but 22rational
3.11.928863111511✓all but 22rational
3.11.928873111511✓all but 22rational
3.11.928903111511✓all but 22rational
3.11.928913111511✓all but 22rational
3.11.928923111511✓all but 22rational
3.11.928933111531✓all but 22rational
3.11.928973111521✓all but 22rational
3.11.928983111521✓all but 22rational
3.11.929063111511✓all2rational
3.11.929073111511✓all2rational
3.11.929083111521✓all2rational
3.11.929093111511✓all2rational
3.11.929103111511✓all2rational
3.11.929123111511✓all2rational
3.11.929133111501✓all2rational
3.11.929143111501✓all2rational
3.11.929153111502✓all2rational
3.11.929173111532✓all2rational
3.11.929193111522✓all2rational
3.11.929203111522✓all2rational
3.11.929213111522✓all2rational
3.11.929223111521✓all2rational
3.11.929253111521✓all2rational
3.11.929283111521✓all2rational
3.11.929293111501✓all2rational
3.11.929303111521✓all2rational
3.11.929313111511✓all but 22rational
3.11.929413111521✓all2rational
3.11.929423111511✓all2rational
3.11.929443111511✓all2rational
3.11.929453111511✓all2rational
3.11.929473111511✓all2rational
3.11.929483111511✓all2rational
3.11.929493111511✓all2rational
3.11.929503111511✓all2rational
3.11.929523111511✓all2rational
3.11.929543111501✓all2rational
3.11.929553111501✓all2rational
3.11.929563111501✓all2genus 1 curve (14a4)
3.11.929573111511✓all2rational

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