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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 20,401–20,450

label ▲rank ⇅n ⇅#bases ⇅#automorphisms ⇅simple ⇅char set ⇅rs dim ⇅birational type ⇅
3.11.913123111501✓all2rational
3.11.913133111501✓all2rational
3.11.913143111511✓all2rational
3.11.913153111501✓all2rational
3.11.913163111501✓all2rational
3.11.913203111491✓all2point over Q(a)/(a^2 - a + 2)
3.11.913233111492✓all2empty
3.11.913283111491✓all2empty
3.11.913293111501✓all2rational
3.11.913303111491✓all2point over Q(a)/(a^4 - a^3 + a^2 - a + 1)
3.11.913313111501✓all2rational
3.11.913323111491✓all2point over Q(a)/(a^2 + 1)
3.11.913393111491✓all2point over Q(a)/(a^2 + 1)
3.11.913443111511✓all2rational
3.11.913453111501✓all2rational
3.11.913463111501✓all2rational
3.11.913473111541✓all2rational
3.11.913513111531✓all2rational
3.11.913523111531✓all2rational
3.11.913553111531✓all2rational
3.11.913583111521✓all2rational
3.11.913633111521✓all2rational
3.11.913703111521✓all2rational
3.11.913753111521✓all2rational
3.11.913843111521✓all2rational
3.11.914003111511✓all2empty
3.11.914013111511✓all2empty
3.11.914023111511✓all2empty
3.11.914043111511✓all2rational
3.11.914053111511✓all2rational
3.11.914063111511✓all2rational
3.11.914083111511✓all2rational
3.11.914093111511✓all2rational
3.11.914103111511✓all2rational
3.11.914113111531✓all2rational
3.11.914133111521✓all2rational
3.11.914143111521✓all2rational
3.11.914153111521✓all2rational
3.11.914163111521✓all2rational
3.11.914183111511✓all2rational
3.11.914193111511✓all2rational
3.11.914203111511✓all2rational
3.11.914213111521✓all2rational
3.11.914223111511✓all2rational
3.11.914253111512✓all2rational
3.11.914303111521✓all2rational
3.11.914313111521✓all2rational
3.11.914323111521✓all2rational
3.11.914353111511✓all2rational
3.11.914383111521✓all2rational

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