Let $p\ge 5$ be prime. For every $j\in\mathbb{F}\_p$, there are at most 6 twists of any elliptic curve over $\mathbb{F}\_p$ with $j$-invariant $j$, and in general only two twists.


Is there a formula which gives the point counts $\#E(\mathbb{F}\_p)$ (equivalently, the traces of Frobenius $a\_p(E)$) of the finitely many elliptic curves over $\mathbb{F}\_p$?


In particular, if $j\in\mathbb{F}\_p$ and $j\not\equiv 0,1728\mod p$, then there are precisely two nonisomorphic elliptic curves $E\_j,E\_j'$ over $\mathbb{F}\_p$ with $j$-invariant $j$. Is there a relation between $a\_p(E\_j),a\_p(E\_j')$?



twists of elliptic curves over finite fields
If $p>2$ and $j \neq 0,1728$ then the two twists are of the form $E: y^2=f(x)$ and
$E\_d: d{y}^2=f(x)$ with $d \notin {\mathbb{F}\_p^{\*}}^{2}$, so $a\_{E\_d} = -a\_E$, as follows: Denote $\chi(0) = 0$, $\chi(x) = 1$ if $x \in {\mathbb{F}\_p^{\*}}^{2}$, $\chi(x) = -1$ otherwise. Then $\#{E(\mathbb{F}\_p)} = 1 + \Sigma\_{x \in \mathbb{F}\_p} (1+\chi(f(x)))$ so $a\_E= - \Sigma \chi(f(x))$, $a\_{E\_d}= - \Sigma \chi(f(x)/d) = -a\_E$.


