Odd torsion of elliptic curves are isomorphic $C: Y^2=X(X^2+aX+b)$ 
$D: Y^2=X(X^2+a_1X+b_1)$ 
where $a,b,\in\mathbb Z a_1=-2a,b_1=a^2-4b,b(a^2-4b)\neq0$
Let $C_{oddtors}(\mathbb Q)$ denote the set of torsion elements of $C(\mathbb Q)$ which have odd order and $D_{oddtors}(\mathbb Q)$ denote the set of torsion elements of $D(\mathbb Q)$ which have odd order. Show that $C_{oddtors}(\mathbb Q)$ and $D_{oddtors}(\mathbb Q)$ are isomorphic.
I don't quite know where to start on this? 
I've already done a section on a 2-isogeny on an elliptic curve and I know that this is where I get the two curves from.
I've considered trying to finding the discriminant and perhaps using Nagell-Lutz Theorem to give an idea of what the torsion points could be.
$d_C=b^2(4b-a^2)$ and $d_D=-16b(a^2-4b)^2$ but then how can I purposely restrict to just looking at the odd torsions?
Any hints in the right direction will be appreciated.
Also, does the question implicitly imply that $C_{eventors}(\mathbb Q)$ and $D_{eventors}(\mathbb Q)$ are not necessarily isomorphic?

Let $E$ and $E'$ be elliptic curves, and let $\phi:E\to E'$ be a $p$-isogeny (i.e., $\phi$ is an isogeny of degree $p$), where $p$ is prime. In particular, $\phi$ is a group homomorphism from $E$ to $E'$ and its kernel $\ker(\phi)$ is a group of size $p$. 


*

*Prove that every $P$ in $\ker(\phi)$ has order dividing $p$.

*Prove that the prime-to-$p$ torsion subgroup of $E$ injects into $E'$. Hint: if $P$ is a torsion point of order $n$ with $\gcd(n,p)=1$, and $\phi(P)=\mathcal{O}_{E'}$, the zero of $E'$, then $P\in\ker(\phi)$... So what is $n$?

*Now consider the dual isogeny $\hat{\phi}:E'\to E$ to show that the prime-to-$p$ torsion subgroup of $E'$ injects into $E$.
About your last question, the "even torsion" in two isogenous curves need not be isomorphic. Let $E:y^2=x^3-x$ and $E':y^2=x^3+4x$. Then $E$ and $E'$ are $2$-isogenous, but $E(\mathbb{Q})_\text{tors}\cong \mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ and $E'(\mathbb{Q})_\text{tors}\cong \mathbb{Z}/4\mathbb{Z}$.
