OEIS A006561 gives the number of intersection points in the diagonals of a regular polygon. There's a paper by Poonen. For 4 vertices to 12, the number of intersection points is:
$$1, 5, 13, 35, 49, 126, 161, 330, 301$$
That's not minimal. Here are my best values:
$$1, 5, 13, 29, 49, 99, 157, 225, 301$$
$7$ -- $29$ -- Drop a vertex of a regular octagon.
$9$ -- $99$ -- (shown below)
$10$ -- $157$ -- Drop vertices 1, 4 of a regular dodecagon.
$11$ -- $225$ -- Drop a vertex of a regular dodecagon.
Can anyone improve on these values, or extend them? EDIT: This is OEIS A230281, which is so far known as $1, 5, 13, 29, 49$. Higher values are unknown. a(2m) is not achieved by the regular 2m-gon in the case of a(10).
EDIT: I've improved my result for 9 points.