Let X be an elliptic curve and P the Riemann sphere. Since X is compact, it is a deep theorem of Douady that the set O(X, P) consisting of holomorphic maps X → P admits a complex structure. If R n denotes the set of maps of degree n, then Namba has shown for n ≥ 2 that R n is a 2n-dimensional complex manifold. We study holomorphic flexibility properties of the spaces R 2 and R 3 .Firstly, we show that R 2 is homogeneous and hence an Oka manifold. Secondly, we present our main theorem, that there is a six-sheeted branched covering space of R 3 that is an Oka manifold. It follows that R 3 is C-connected and dominable. We show that R 3 is Oka if and only if P 2 \C is Oka, where C is a cubic curve that is the image of a certain embedding of X into P 2 .We investigate the strong dominability of R 3 and show that if X is not biholomorphic to C/Γ 0 , where Γ 0 is the hexagonal lattice, then R 3 is strongly dominable.As a Lie group, X acts freely on R 3 by precomposition by translations. We show that R 3 is holomorphically convex and that the quotient space R 3 /X is a Stein manifold.We construct an alternative six-sheeted Oka branched covering space of R 3 and prove that it is isomorphic to our first construction in a natural way. This alternative construction gives us an easier way of interpreting the fibres of the branched covering map.The full thesis is available at https://arxiv.org/abs/1609.07184.