New isogenies of elliptic curves over number fields

We analyze the fields of definition of cyclic isogenies on elliptic curves to prove the following uniformity result: for any number field F₀ which satisfies an isogeny condition, there exists a constant B:=B(F₀)⁺ such that for any finite extension L/F₀ whose degree [L:F₀] is coprime to B, one has for all elliptic curves E/F₀ with j-invariant ≠ 0, 1728 that any L-rational cyclic isogeny on E must be F₀-rational. We also prove unconditional results for the mod- Galois representations of non-CM elliptic curves with an F₀-rational -isogeny when is uniformly large.