of $J({\mathbb Q})$ and the ${\mathbb F}_2$-dimensions of the other groups that we defined. $$\label{eq-rank}
{{\rm rank\,}}_{\mathbb Z}J({\mathbb Q}) + \dim_{{\mathbb F}_2} J({\mathbb Q})[2] + \dim_{{\mathbb F}_2}\Sha({\mathbb Q},J)[2] = \dim_{{\mathbb F}_2}{{\rm Sel}}^{(2)}({\mathbb Q},J).$$ Using equation (\[eq-rank\]), we get our first upper bound on the rank $$\label{eq-bound}
{{\rm rank\,}}_{\mathbb Z}J({\mathbb Q})\leq \dim_{{\mathbb F}_2} {{\rm Sel}}^{(2)}({\mathbb Q},J)-\dim_{{\mathbb F}_2} \Sha({\mathbb Q},J)[2]\leq \dim_{{\mathbb F}_2} {{\rm Sel}}^{(2)}({\mathbb Q},J).$$ This upper bound is [*computable*]{}, in the sense that $J({\mathbb Q})[2]$ and the Selmer group are computable, as we describe below.

\[defn-L\] For any field extension $K$ of ${\mathbb Q}$, let $L_K = K[T] / (f(T))$ denote the algebra defined by $f$ and let $N_K$ denote the norm map from $L_K$ down to $K$.

We denote $L_K=K[\theta]$, where $\theta$ is the image of $T$ under the reduction map $K[T]\to K[T]/(f(T))$, and $L_K$ is a product of finite extensions of $K$: $$L_K=L_{K,1}\times\cdots\times L_{K,m_K},$$ where $m_K$ is the number of irreducible factors of $f(x)$ in $K[x]$. Here, the fields $L_{K,j}$ correspond to the irreducible factors of $f(x)$ in $K[x]$, and the map $N_K:L_K\to K$ is just the product of the norms on each component of $L_K$. That is, if $\alpha=(\alpha_1,\alpha_2,\dots,\alpha_{m_K})$, then $N_K(\alpha)=\prod_{i=1}^{m_K}N_{L_{K,i}/K}(\alpha_i)$ where $N_{L_{K,i}/K}:L_{K,i}\to K$ is the usual norm 