  Given a rational point on a curve in a rational isogeny class, a natural
question concerns the field of definition of its pre-images. The multiplication
by m endomorphism is a powerful and much-used tool. The pre-images for this map
are found by factorizing a monic polynomial of degree m^2. For m = 2, Everest
and King gave examples where the existence of a quadratic factor coincided with
the existence of a rational pre-image via a 2-isogeny. Nelson Stephens asked if
this always happens and the question is answered in the affirmative. It is also
shown that the analogue for m = 3 can only be false when there exists a
rational point of order three and a small number of counterexamples are found.
The results are proven over any field with characteristic not two or three.
