On the Jacobian of hyperelliptic curves $y^2=x^5+m^2$

In this paper, we study the algebraic rank and the analytic rank of the Jacobian of hyperelliptic curves $y^2=x^5+m^2$ for integers $m$. Namely, we first provide a condition on $m$ that gives a bound of the size of Selmer group and then we provide a condition on $m$ that makes $L$-functions non-vanishing. As a consequence, we construct a Jacobian that satisfies the rank part of the Birch–Swinnerton-Dyer conjecture.