organization of the paper is as follows. In Section \[sec-2descent\], we review the method of $2$-descent as implemented by Stoll in [@stoll]. In Sections \[sec-betterbounds\], \[sec-totally\], and \[sec-muchbetterbounds\], we specialize the $2$-descent method for the situations we encounter in the rest of the paper, namely the case when $f(x)$ defines a totally real extensions, or cyclic extension of ${\mathbb Q}$, of prime degree. In Section \[sec-genus1\], we give a new proof of Washington’s theorem using the method of $2$-descent. In Section \[sec-sophie\] we provide examples of hyperelliptic curves of genus $g=(p-1)/2$ where $p$ is a Sophie Germain prime, and prove Theorem \[thm-sophie-intro\]. Finally, in Section \[sec-examples\], we illustrate the previous sections with examples of curves, jacobians, and how their ranks compare to the bounds.

The authors would like to thank Keith Conrad, Gürkan Dogan, Franz Lemmermeyer, Paul Pollack, and Barry Smith, for several helpful comments and suggestions. We would also like to express our gratitude to David Dummit for very useful suggestions and noticing an error in an earlier version of the paper.

Stoll’s Implementation of 2-Descent {#sec-2descent}
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In this section we summarize the method of $2$-descent as implemented by Stoll in [@stoll]. The method was first described by 