elliptic curves - Q-isogeny and Q-torsion subgroup

I assume you have an elliptic curve $E$ defined over $mathbb{Q}$ -- in simplest terms, this means a Weierstrass equation $y^2 = x^3 + Ax + B$ with $A,B in mathbb{Q}$.

Then a $mathbb{Q}$-isogeny is a finite morphism $varphi: (E,O) rightarrow (E',O')$ which is defined over $mathbb{Q}$: in other words, given locally by rational functions with $mathbb{Q}$-coefficients. An equivalent perspective is that an isogeny is essentially determined -- i.e., up to an automorphism on the target -- by its kernel $E[phi]$, a finite
subgroup of $E(overline{mathbb{Q}})$. Then the definedness over $mathbb{Q}$ is equivalent to invariance under the group $operatorname{Aut}(overline{mathbb{Q}}/mathbb{Q})$: for every Galois automorphism $sigma$, we want $sigma E[phi] = E[phi]$.

Similarly, the $mathbb{Q}$-torsion subgroup is the subgroup of the full torsion subgroup which is defined over $mathbb{Q}$. This means $E[operatorname{tors}] cap E(mathbb{Q})$: it is just the subgroup of $E(mathbb{Q})$ consisting of points of finite order. It can also (equivalently) be defined as the Galois invariants of $E[operatorname{tors}](overline{mathbb{Q}})$ (= $E[operatorname{tors}](mathbb{C})$).

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