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This master thesis describes how Selmer groups can be used to determine theMordell-Weil group of elliptic curves over a number field K. The Mordell-WeilTheorem states that $E(K) = E(K)_{tors} \times Z^r$, where $r$ is the rank of$E$, and $E(K)_{tors}$ is the torsion subgroup, i.e. the group of points offinite order in $E(K)$. The group $E(K)_{tors}$ is finite and well understood.So, one tries to find a way to determine the rank $r$ of $E$, which is themajor problem. The procedure described in this thesis shows how to transfer thecomputation of the weak Mordell-Weil group $E(K)/mE(K)$ to the existence ornon-existence of a rational point on certain curves, called homogeneous spaces.If one can find some completion $K_v$ of $K$ such that the homogeneous spacehas no points in $K_v$, then it follows that it has no points in $K$. Under theassumption that the Shafarevich-Tate group is finite, the rank of Ellipticcurves over $\mathbb{Q}$ with $j$-invariant $1728$ is fully determined incertain cases.

Using Selmer Groups to compute Mordell-Weil Groups of Elliptic Curves