What is meant by a Qisogeny and the Qtorsion subgroup? (And by Q, I mean rational 'Q')`
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What is meant by a Qisogeny and the Qtorsion subgroup? (And by Q, I mean rational 'Q')` ` 


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 $E[\operatorname{tors}](\overline{\mathbb{Q}})$ (= $E[\operatorname{tors}](\mathbb{C})$). 

