The question:
Let C be a smooth curve of genus one defined over $K$.
(a)Prove that $j(C)\in K$.
(b)Prove that C is an elliptic curve over $K$ if and only if C$(K)\neq \emptyset$.
(c)Prove that C is always isomorphic over $\overline K$ to an elliptic curve defined over $K$.
Part (b) follows from the definition of elliptic curves and the fact that C is of genus one.
For part (c),since C is of genus one it is isomorphic to an elliptic curve,say $E$,over $\overline K$.Now,assuming (a), $j$(C)$\in K$ implies that there is an elliptic curve $E'$ over $K(j$(C))=$K$.And since two elliptic curves with same $j$-invariant are isomorphic,it shows that,C is infact isomorphic to $E'$ which is an elliptic curve defined over $K$.This proves (c).
Is the proof correct?And how to go about proving part (a)?If C$(K)\neq \emptyset$ then C is an elliptic curve over $K$,and hence it is given by a Weierstrass equation with co-efficients in $K$.The general formula for $j$-invariant shows that it is in $K$.But what if C$(K)=\emptyset$?
Does it follow directly from the general Weierstrass equations.