  We show that for all epsilon > 0, there is a constant C(epsilon) > 0 such
that for all elliptic curves E defined over a number field F with j(E) in Q we
have #E(F)[tors] \leq C(epsilon)[F:Q]^{5/2+epsilon}. We pursue further bounds
on the size of the torsion subgroup of an elliptic curve over a number field
E/F that are polynomial in [F:Q] under restrictions on j(E). We give an
unconditional result for j(E) lying in a fixed quadratic field that is not
imaginary of class number one as well as two further results, one conditional
on GRH and one conditional on the strong boundedness of isogenies of prime
degree for non-CM elliptic curves.
