How to compute $S^{(2)}(E / \mathbb{Q}))$ from $S^{(\phi)} (E/ \mathbb{Q})$ and $S^{(\hat \phi)} (E'/ \mathbb{Q})$? I'm looking at example 4.10 (From chapter X) of Silverman's book "The Arithmetic of Elliptic Curves" 2nd Edition, which is about Descent via two-isogeny (proposition 4.9). The example is about computing $E(\mathbb{Q}) / 2 E(\mathbb{Q})$ for the curve
$$ E: y^2 = x^3 - 6x^2 +17x .$$
The curve that is $2$-isogenous to $E$ is
$$ E' : Y^2 = X^3 + 12X^2 - 32X .$$
We have the usual isogeny $\phi: E \to E'$ and dual isogeny $\hat \phi : E' \to E$ with $\hat \phi \circ \phi = 2$ (the exact formulas are defined in proposition 4.9).
He computes that
$S^{(\phi)} (E/ \mathbb{Q}) = \{ \pm 1, \pm 2\}$
and
$S^{(\hat \phi)} (E'/ \mathbb{Q}) = \{ 1, 17 \}.$
I would like to compute the $2$-Selmer group $S^{(2)}(E / \mathbb{Q})$, and from what I understand I need to combine the above two information somehow (similar to how you can combine $E'(\mathbb{Q}) / \phi ( E(\mathbb{Q}))$ and $E(\mathbb{Q}) / \hat \phi (E' (\mathbb{Q}))$ to get $E(\mathbb{Q}) / 2E(\mathbb{Q}))$. Could anyone explain how I can go about doing this?
P.S. I feel pretty overwhelmed by this book in general, so if I've gotten it all wrong please let me know.
 A: In your example, the revelant information you need is remark X.4.7 and the fact that $$E'(\mathbb{Q}) / \phi ( E(\mathbb{Q}))=S^{(\phi)} (E/ \mathbb{Q})=\{ \pm 1, \pm 2\} \text{ and } E(\mathbb{Q}) / \hat \phi (E' (\mathbb{Q}))=S^{(\hat \phi)} (E'/ \mathbb{Q})=\{ 1, 17 \},$$
since the relevant Tate-Shafarevich groups are trivial, the identification via "the $x$-coordinate" (except for the two torsion point $(0,0)$).
So we have remark X.4.7  saying that
$$ E'(\mathbb Q)[\hat \phi]/\phi(E(\mathbb Q)[2])\to E'(\mathbb Q)/\phi(E(\mathbb Q)) \to E(\mathbb Q)/2E(\mathbb Q) \to E(\mathbb Q)/\hat \phi (E'(\mathbb Q)) \to 0$$
is exact, the second morphism being $\phi$.
Now, $$E'(\mathbb Q)/\hat \phi(E(\mathbb Q)) =\langle\ \overline{(0,0)}\ , \ \overline{(8,-32)}\ \rangle$$
and the map $\phi$ sends $(0,0)$ to $0$ (the point at infinity) (this is of course general, since $\phi$ is the 2-isogeny sending $(0,0)$ to $0$), and sends $(8,-32)$ to an element $\phi(8,-32)$, not $2$-torsion (which I left to you to compute).
So $$S^{(2)}(E/\mathbb Q)=E(\mathbb Q)/2E(\mathbb Q) =\langle\  \overline{\phi(8,-32)}\ , \ \overline{(0,0)}\ \rangle.$$
Now you can use the description you like of the elements of $S^{(2)}(E/\mathbb Q)$.
Edit: In the general case, there exists a natural exact sequence
$$\to S^{(\phi)} (E/ \mathbb{Q}) \to S^{(2)}(E/\mathbb Q) \to S^{(\hat \phi)} (E'/ \mathbb{Q}) \to 0$$
but I am not sure about what to put in the left hand side.
One possible way to understand this sequence is to observe that $S^{(2)}(E/\mathbb Q)$ is the group of unramified Galois coverings of $E$ with Galois group "$E[2]$" (and with points locally everywhere), and the 2-isogeny comes from a $\mathbb Z/2\mathbb Z \subset E[2]$: the map  $ S^{(2)}(E/\mathbb Q) \to S^{(\hat \phi)} (E'/ \mathbb{Q}) $ correspons to the map "fixing by  the $\mathbb Z/2\mathbb Z$", like in the usual Galois theory of fields.
The map $S^{(\phi)} (E/ \mathbb{Q}) \to S^{(2)}(E/\mathbb Q)$ corresponds to see each unramified Galois covering $\psi:D\to E'$ of $E'$ with Galois group $\mathbb Z/2\mathbb Z$ to a Galois covering of $E$ via $\phi$, i.e. $\phi\circ \psi:D\to E'\to E$.
