= Solution
The $q$-power <Frobenius isogeny> sends affine coordinates to $(x^q,y^q)$, has degree $q$, and fixes exactly the points defined over $\mathbb F_q$, including the origin. Consequently
$$
E(\mathbb F_q)=\ker(1-\phi)(\overline{\mathbb F}_q).
$$
The degree is $q$ because the $q$-power Frobenius on a smooth curve has function-field inseparable degree $q$. By the permitted separability of $1-\phi$, this <separable isogeny> has exactly $\deg(1-\phi)$ geometric kernel points; there is no inseparable multiplicity to remove. Part(i) therefore gives
$$
\boxed{\#E(\mathbb F_q)=\deg(1-\phi)=q+1-\operatorname{tr}(\phi).}
$$
This relates the <elliptic-curve point count over a finite field> to the integer trace, rather than treating the point count as an independently assumed definition of the trace.
Back to article page
