Section 16.4

It makes sense that for computer usage that having elliptic curves mod 2 would be worthwhile, what i hadnt thought of is that the equations would need to be modified to not make them = infinity everywhere.  If we are finding the points on a line though mod 2 would there ever be more than 5 points total?  It seems like no matter the curve that they would always only be able to have the same 5 points.  I guess that that is probably why it says that usually elliptic curves mod 2 are not generally big enough.