  Let $E/\mathbb{Q}$ be an elliptic curve of conductor $N$ and let $K$ be an
imaginary quadratic field of discriminant prime to $N$. The choice of $K$
induces a decomposition $N=N^+N^-$ where every prime $\ell|N^+$ splits in $K$
and every prime $\ell|N^-$ is inert in $K$. Originally assuming $N^-=1$,
Kolyvagin constructed cohomology classes for $E$ using $K$-CM points and
conjectured they did not all vanish. Conditional on this conjecture, he
described the Selmer rank of $E$ using his system of classes. When $N^-$ is the
squarefree product of an even number of primes, we extend work of Wei Zhang to
prove new cases of Kolyvagin's conjecture. We also prove an analogous result,
and give a description of the Selmer group, when $N^-$ is the squarefree
product of an odd number of primes.
