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4 1 $begingroup$ This is taken from The Arithmetic of Elliptic Curves by Silverman on page 154, Q5.10(f). One way of directly solving this problem is to work out on sage all 8^5 possibilities of elliptic curves and show that no such curve with the required number of points exist. This has been done and in that sense, the problem has been solved. However, the question advises to use the previous part, which states: Let $p^i$ be the largest power of $p$ such that $p^{2i}|q$. Then $tr(phi)=0mod p iff tr(phi)=0mod p^i$. For the case of part (f), this does not seem to be at all useful as $p=2$ and $q=8$, which implies that $i=1$ and so the previous part does not yield any new information. It might be the case that there is a typo but by testing for several cases of $q$, this was found to not be the case. Is