Way to find all the $j$-invariants for elliptic curves over a field

Question

We can group elliptic curves into isomorphism classes by their $j$-invariants, but is there a way to know how many and what are the valid $j$-invariants for elliptic curves over a fixed finite field?

Answer 1

As Angina says, for every $a\in k$ there is an elliptic curve $E/k$ with $j(E)=a$. If $\operatorname{char}(k)\ne2,3$ and $a\ne0,1728$, then for a fixed $a$, the $k$-isomorphism classes of elliptic curves with $j(E)=a$ are in 1-to-1 correspondence with the quotient group $k^*/{k^*}^2$. If $k$ is a finite field, then this group has order $2$. For $a=0$ or 1728, still assuming $\operatorname{char}(k)\ne2,3$, the isomorphism classes are in 1-to-1 correspondence with $k^*/{k^*}^6$, respectively $k^*/{k^*}^4$. So in those cases, for finite fields, the number of isomorphism classes depend on the characteristic mod 12. For $\operatorname{char}(k)=2,3$ it gets more complicated.