  This article deals with the Galois representation attached to elliptic curves
with an isogeny of prime degree over a number field. We first determine uniform
criteria for the irreducibility of Galois representations attached to elliptic
curves in some infinite families, characterised by their reduction type at some
fixed places of the base field. Then, we give an explicit form for a bound that
appear in a theorem of Momose. Finally, we use these results to precise a
previous theorem of the author about the homotheties contained in the image of
the Galois representation.
