Does this equation have a solution in $\mathbb{Q}_{11}$ and in $\mathbb{Q}_p$?

Question

In the department where I am studying there was a talk on Hilbert Symbols and Hasse principle.

The orator asked us the following question on which I am not able to make any progress. So, I am asking for help here.

Question : Does $f: 7 x^2 + 4y^2 =z^2 $ have a solution in (i) $\mathbb{Q}_{11}$ and in (ii) $\mathbb{Q}_p$?

Attempt: I am not able to make any progress on this question. I don't think there is a point using hit and trial method here. I have not studied any algebraic number theory but I thought I should try the question. My background in Algebra is strong.

Can you please let me know how should I approach the solution? Which results should I use?

Thanks!

Answer 1

COMMENT.-Because of the diophantine equation $f(x)=7x^2+4y^2-z^2=0$ has the integer solution $(x,y,z)=(2,3,8)$, we can say (because $\mathbb Q$ is embedding in $\mathbb Q_p$) that $f(x)=0$ has solution for all prime $p$. However I understand the purpose of this problem is giving an exercise of verification for a particular example of Hasse principle. It is, in fact, mainly an exercise of working in $\mathbb Q_p$