L-tables, unfolded into Swiss crosses, give genus 2 surfaces with a single cone point. The following pictures all come from surfaces with lattice Veech groups (by the McMullen/Calta classification); the corresponding L-table comes from a square with side length a = (1+sqrt(d))/2, from which a corner square has been cut, leaving two sides of length 1.
d=3
d=5
d=6
d=7
d=10
d=11
d=13
d=14
d=15
d=17
d=19
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