Rank bound for a prime congruent-number elliptic curve ($\mathrm{rank}{E}_p(\mathbb{Q})\le 2$)

For an odd prime $p$, let $E_p:y^2=x^3-p^{2}x$. Its two-isogeny descent partner is $E_p^{\prime }:y^2=x^3+4p^{2}x$. 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 $E_p^{\prime }$ has positive $x$, so its square-class image is contained in $\{1,2,p,2p\}$. The two-isogeny rank formula gives $2^{\mathrm{rank}{E}_p}=\#{\alpha }^{\prime }(E_p^{\prime })\le 4$, hence the rank is at most $2$.