Rank-zero elliptic curve with cubic x cubed plus x ($y^2=x^3+x,r=0$)

For $E:y^2=x^3+x$, the two-torsion square-class homomorphism has image $\{1\}$: prime-support bounds allow only $\pm 1$, and every nonzero affine $x$ is positive. Its two-isogeny formula companion $E^{\prime }:y^2=x^3-4x$ has possible image classes $\{1,-1,2,-2\}$, all realized by its rational two-torsion. The two-isogeny rank formula gives $2^r=1\cdot 4/4=1$, so the rank is zero.