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 $E(\Bbb F_5)\cong C_{10}$.


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