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At every good prime we have $p\ne 2$. After removing any even p-adic valuation in the parameter as in part (a), use a smooth reduction $y^2=x^3-d^{2}x$ with $d\in {\mathbb{F}}_p^{\times }$. Let $\chi$ be the Legendre symbol, extended by $\chi (0)=0$. Counting the two, one or zero y-values above each x givesIf $p\equiv 3(\mathrm{mod}4)$, then $\chi (-1)=-1$. The terms for $x$ and $-x$ cancel, including zeros, because the cubic is odd. Hence $a_p=0$.

For the converse, the four distinct points $O,(0,0),(d,0),(-d,0)$ form the full rational 2-torsion subgroup of the reduced curve. Its group order is therefore divisible by four. If $p\equiv 1(\mathrm{mod}4)$, zero trace would give group order $p+1\equiv 2(\mathrm{mod}4)$, a contradiction. ConsequentlyThis is the vanishing trace criterion for a congruent number curve. The Hasse theorem for elliptic curves statesFor good primes that divide a nonsquarefree displayed parameter, the tilde denotes the smooth reduction of a minimal model, not the singular special fibre of that nonminimal equation.