  Let $X$ be a minuscule homogeneous space, an odd quadric, or an adjoint
homogenous space of type different from $A$ and $G_2$. Le $C$ be an elliptic
curve. In this paper, we prove that for $d$ large enough, the scheme of degree
$d$ morphisms from $C$ to $X$ is irreducible, giving an explicit lower bound
for $d$ which is optimal in many cases.
