is very close to 1728. Viewed in another light, then, this result says that j(EA,B) cannot be “too close” to 1728 (relative to the sizes of A and B) if EA,B(Q) contains a torsion point of order at least 5. In [5], the author proved similar results bounding |B| by a power of |A|, again for EA,B with certain torsion/isogeny structure. One may similarly view these results as saying that j(EA,B) cannot be “too close” to 0 for curves EA,B with given torsion structure. This prompts us to formulate here a general result on the approximation of a fixed algebraic number by the j-invariants of elliptic curves with certain torsion structure. The main results are stated over arbitrary number fields and contain, as special cases, most of the results in [2, 5]. It is easy to show that for each N , {j(E) : E/Q admits a rational N -isogeny}