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We unconditionally determine $I_\Q(d)$, the set of possible prime degrees ofcyclic $K$-isogneies of elliptic curves with $\Q$-rational $j$-invariants andwithout complex multiplication over number fields $K$ of degree $\leq d$, for$d\leq 7$, and give an upper bound for $I_\Q(d)$ for $d>7$. Assuming Serre'suniformity conjecture, we determine $I_\Q(d)$ exactly for all positive integers$d$.

Isogenies of non-CM elliptic curves with rational $j$-invariants over number fields