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 $a_p(E_j),a_p(E_j')$?