Rank-one elliptic curve with cubic x cubed minus twice x ($y^2=x^3-2x,r=1$)

For $E:y^2=x^3-2x$, prime-support bounds allow image classes $\{1,-1,2,-2\}$. The rational points $(0,0)$ and $(2,2)$ realize $-2$ and $2$, and the two-torsion square-class homomorphism is multiplicative, so all four occur. Its companion $E^{\prime }:y^2=x^3+8x$ has only positive candidate classes $1,2$, both realized by the identity and $(0,0)$. The two-isogeny rank formula gives $2^r=4\cdot 2/4=2$, so the rank is one. The point $(2,2)$ has infinite order: its double is $(9/4,-21/8)$, contradicting the integrality of torsion coordinates in the Nagell–Lutz theorem if it were torsion.