  Let E be an elliptic curve over the rationals. Let L be an infinite Galois
extension of the rationals with uniformly bounded local degrees at almost all
primes. We will consider the infinite extension L(E_tor) of the rationals where
we adjoin all coordinates of torsion points of E. In this paper we will prove
an effective lower bound for the height of non-zero elements in L(E_tor) that
are not a root of unity.
