= Rank bound for a prime congruent-number elliptic curve
{title2=$\operatorname{rank}E_p(\mathbb Q)\le2$}
For an odd <prime> $p$, let $E_p:y^2=x^3-p^2x$. Its <two-isogeny descent> partner is $E'_p:y^2=x^3+4p^2x$. The first <two-torsion square-class homomorphism> has image exactly $\{1,-1,p,-p\}$: prime-support bounds allow only these classes, and the four rational two-torsion points realize all of them. Every nonidentity point on $E'_p$ has positive $x$, so its square-class image is contained in $\{1,2,p,2p\}$. The <two-isogeny rank formula> gives $2^{\operatorname{rank}E_p}=\#\alpha'(E'_p)\le4$, hence the rank is at most $2$.
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