# Isomorphism between elliptic curves over $\mathbb Q$ and $\mathbb F_5$

Given are the Elliptic curves $$E_1 : y^2 = x^3+x$$ and $$E_2 = y^2 = x^3+3x$$. Are these isomorphic over

a) $$\mathbb Q$$?

b) $$\mathbb F_5$$?

I see they are isomorphic over $$\mathbb C$$, as they have the same $$j$$-invariant. I suppose they aren't isomorphic, but how to prove this?

Hint: They are isomorphic over $$K$$ if and only there exists a $$u\in K^{\times}$$ such that $$u^4=3$$, see here.
Actually, for $$y^2=x^3+x$$ we have $$E(\Bbb F_5)\cong C_2\times C_2$$, and for $$y^2=x^3+3x$$ we have $$E(\Bbb F_5)\cong C_{10}$$.