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In this paper, we show that the Birch and Swinnerton-Dyer conjecture for a certain elliptic curve over    $$\mathbb {Q}\left( \root 4 \of {5}\,\right) $$    Q    5  4is equivalent to the same conjecture for a certain pair of hyperelliptic curves of genus 2 over    $$\mathbb {Q}$$   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.

The Birch and Swinnerton-Dyer conjecture for an elliptic curve over $$\pmb {\mathbb {Q}\left( \root 4 \of {5}\,\right) }$$