Consider the equation $$(6-\sqrt{37})y^2=(6-\sqrt{37})^2-68x^4 $$ over $\Bbb{Q}_2(\sqrt{37})=\Bbb{Q}_2(\sqrt{5})$ and over $\Bbb{Q}_{17}(\sqrt{37})$.
I guess from computational caluculation that it does not have solution over either $\Bbb{Q}_2(\sqrt{5})$ or $\Bbb{Q}_{17}(\sqrt{37})$. But I'm unable to prove that by hand. It seems extremely difficult for me.
For example, suppose the equation has a solution in $\Bbb{Q}_2(\sqrt{5})$. Taking the 2-adic valuation $v$ gives $2v(y)=\min\{0,2+4v(x)\}$. But I couldn't find any contradiction.
Thank you for your help.