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The class number divisibility problem for number fields is one of theclassical problems in algebraic number theory, which originated from Gauss'class number conjectures. The relation between the points on an elliptic curveand class number divisibility of a number field has been explored through theworks of various mathematicians. Here, we explicitly construct an unramifiedabelian extension of a bi-quadratic field generated from points of a certaintype of elliptic curve. Moreover, showing the $2$-Selmer rank of the saidelliptic curve as $1$, we also construct an infinite family of bi-quadraticfields of even class number.

Class number divisibility of $\mathbb{Q}(\sqrt{3p}, \sqrt{m-21pn^{2}})$ constructed from elliptic curves of $2$-Selmer rank exactly $1$