Is there a way to find $k$ elliptic curves over $F_q$, all with provably different number of points?
And doing so without counting the number of points on the curves.
For really small $k$ (2,3) I know some examples of families of curves that has different number of points for different members. But I don't know of a family where this can be achieve for a large k.
I also know that having different j-invariant doesn't ensure different number of points. But maybe there is another way to use the j-invariant here?