= Rank-one elliptic curve with cubic x cubed minus twice x
{title2=$y^2=x^3-2x,\quad r=1$}
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 $E':y^2=x^3+8x$ has only positive candidate classes $1,2$, both realized by the identity and $(0,0)$. The <two-isogeny rank formula> gives $2^r=4\cdot2/4=2$, 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.
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