Let E be an elliptic curve over the rationals without complex multiplication.
The absolute Galois group of Q acts on the group of torsion points of E, and
this action can be expressed in terms of a Galois representation
rho_E:Gal(Qbar/Q) \to GL_2(Zhat). A renowned theorem of Serre says that the
image of rho_E is open, and hence has finite index, in GL_2(Zhat). We give the
first general bounds of this index in terms of basic invariants of E. For
example, the index can be bounded by a polynomial function of the logarithmic
height of the j-invariant of E.