  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.
