We study Heegner points and Kolyvagin classes for elliptic curves
over Q, with special focus on curves that have analytic rank
at least 2. We reinterpret Kolyvagin's ``derived classes''
construction, in the context of divisors on modular curves directly
in characteristic ell, and prove compatibility and multiplicity one
results. We use these results to give the first complete algorithm
for explicitly computing (certain) Kolyvagin classes, and thus
verify a conjecture of Kolyvagin for some specific elliptic curves.