In a rational function the divisors is defined as
$$div(f) = \sum e_i[a_i] - f_i[b_i]$$ for function $$f = \frac{(x-a_i)^e}{(x-b_i)^f}$$
Hence it is the zeros of the numerator and the denominator.
For divisors on f defined on elliptic curve E with point P
$$div_{P\in E}(f) = \sum n_p[P]$$ Where n_p is the order of point P. Here is point P
- the intersection of f and E or
- zeros of f that also lie on E?
I've worked through multiple textbooks/lecture notes and can't seem to figure this out.