and, therefore, $$J({\mathbb Q})\cong J({\mathbb Q})_\text{tors} \oplus {\mathbb Z}^{R_{J/{\mathbb Q}}},$$ where $J({\mathbb Q})_\text{tors}$ is the (finite) subgroup of torsion elements, and $R_{J/{\mathbb Q}}=\operatorname{rank}_{\mathbb Z}(J({\mathbb Q}))\geq 0$ is the rank of $J({\mathbb Q})$. In this article we are interested in bounds of $R_{J/{\mathbb Q}}$ in terms of invariants of $C$ or $f(x)$.

In [@Wash1], Washington showed the following bound for the rank of certain elliptic curves, building on work of Shanks on the so-called simplest cubic fields (see [@Shanks1]).

\[thm-wash\] Let $m\geq 0$ be an integer such that $m^2+3m+9$ is square-free. Let $E_m$ be the elliptic curve given by the Weierstrass equation $$E_m\colon y^2 = f_m(x)= x^3+mx^2-(m+3)x+1.$$ Let $L_m$ be the number field generated by a root of $f_m(x)$, let $\operatorname{Cl}(L_m)$ be its class group, and let ${{\rm Cl}}(L_m)[2]$ be the $2$-torsion subgroup of ${{\rm Cl}}(L_m)$. Then, $$\operatorname{rank}_{\mathbb Z}(E_m({\mathbb Q})) \leq 1 + \operatorname{dim}_{{\mathbb F}_2}(\operatorname{Cl}(L_m)[2]).$$

In this article, we extend Washington’s result to curves of genus $g\geq 2$. In order to find other families of hyperelliptic curves of genus $g\geq 2$ where a similar bound applies, we use a method of $2$-descent for jacobians described by Cassels, Poonen, Schaefer, and Stoll (see Section \[sec-2descent\]; in particular, we follow the implementation described 