Descent by 3-isogeny on elliptic curves

Descent by a rational isogeny has shown to be a useful tool in computing the rank of elliptic curves. After outlining the general theory, we recall the well-known theory of descent by 2-isogeny. A relatively new approach is descent by a 3-isogeny. In this thesis, we formulate an algorithm that computes the Selmer group of any rational 3-isogeny. We apply these techniques to a family of elliptic curves that allow both types of descent. We force elements into the Selmer group of the 3-isogeny, while we show by 2-descent that the elliptic curves have rank zero. This yields a construction of elements of order 3 in a Tate-Shafarevich group.