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Assume tha $C$ be the hyperelliptic curve $y^2 = (x-a_1)\cdots (x-a_5)$ of genus $g=2$ and $a_i \in \mathbb{Z}$ and we know that the integers $a_i$ has the form $a_i= d_1^2 - d_i^2$ for some positive integers $d_i$. I am wondering that the Jacobian of the hyperelliptic curve of the above type has rank at most one?Generally how one can find conditions on integral roots $a_i$ which guarantee that the rank of the Jacobian is at most one?

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    $\begingroup$ There are examples of tuples $(d_1,d_2,\ldots,d_5)$ for which the rank of the Jacobian of the corresponding hyperelliptic curve is at least 2. E.g. $(d_1,d_2,\ldots,d_5)=(1,2,3,4,7),(1,2,3,4,8),(1,2,3,4,9),(1,2,3,4,10),(1,2,3,4,11).$ An example when the rank is at least 3 is given by the tuple $(1,2,3,4,29).$ $\endgroup$ – castor Aug 28 '15 at 19:10

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