  In this paper we show the Birch and Swinnerton-Dyer conjecture for a certain
elliptic curve over $\mathbb{Q}(\sqrt[4]{5})$ is equivalent to the same
conjecture for a certain pair of hyperelliptic curves of genus 2 over
$\mathbb{Q}$. We numerically verify the conjecture for these hyperelliptic
curves. Moreover, we explain the methods used to find this example, which
turned out to be a bit more subtle than expected.
