$v=2$ and $d_K = 0$ if $v$ is odd. Then

1.  $\dim J(K)/2J(K) = \dim J(K)[2] + d_Kg =m_K-1+d_Kg.$

2.  $\dim H_K = 2\dim J(K)/2J(K) = 2(m_K-1+d_Kg)$.

3.  $\dim I_K = m_K - 1.$

With all of this machinery the description of ${{\rm Sel}}^{(2)}({\mathbb Q},J)$ given in Proposition \[prop-selmer\_characterization\] can be refined as follows.

\[prop-selmer\_recharacterization\] Let $S = \{\infty,2\}\cup\{v : v^2 \hbox{ divides } {{\rm disc}}(f) \}$. Then $${{\rm Sel}}^{(2)}({\mathbb Q},J) = \{ \xi\in H \mid {{\rm val}}(\xi)\in I_S(L)/I_S(L)^2,\  {{\rm res}}_v(\xi)\in \delta_v(J({\mathbb Q}_v))\hbox{ for all } v\in S\}.$$

This new characterization suggests the following method to compute ${{\rm Sel}}^{(2)}({\mathbb Q},J)$:

-   Find the set $S$.

-   For each $v\in S$, determine $J_v = \delta_v(J({\mathbb Q}_v))\subseteq H_v.$

-   Find a basis for a suitable finite subgroup $\widetilde{H} \subseteq L^\times/(L^\times)^2$ such that ${{\rm Sel}}^{(2)}({\mathbb Q},J)\subseteq \widetilde{H}.$

-   Compute ${{\rm Sel}}^{(2)}({\mathbb Q},J)$ as the inverse image of $\prod_{v\in S} J_v$ under $$\prod_{v\in S} {{\rm res}}_v\colon \widetilde{H}\to \prod_{v\in S} H_v.$$

Ignoring any complications that arise from computing and factoring the discriminant of $f$, we focus on steps 2 and 3. We omit the details of how to carry out step 4, since we are only interested in an upper bound for the ${\mathbb F}_2$-dimension 