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We consider the Kolyvagin cohomology classes associated to an elliptic curve$E$ defined over $\mathbb{Q}$ from a computational point of view. We explainhow to go from a model of a class as an element of$(E(L)/pE(L))^{\mathrm{Gal}(L/\mathbb{Q})}$, where $p$ is prime and $L$ is adihedral extension of $\mathbb{Q}$ of degree $2p$, to a geometric model as agenus one curve embedded in $\mathbb{P}^{p-1}$. We adapt the existing methodsto compute Heegner points to our situation, and explicitly compute them aselements of $E(L)$. Finally, we compute explicit equations for several genusone curves that represent non-trivial elements of the p-torsion part of theTate-Shafarevich group of $E$, for $p \leq 11$, and hence are counterexamplesto the Hasse principle.

Explicit realization of elements of the Tate-Shafarevich group constructed from Kolyvagin classes