= Rank-two elliptic curve with cubic x cubed minus seventeen x
{title2=$E:y^2=x^3-17x,\quad r=2$}
The <two-torsion square-class homomorphism> on $E$ has candidate classes $\{1,-1,17,-17\}$ by the <prime-support bound in two-isogeny descent>. The identity, $(0,0)$ and $(-1,4)$ realize $1,-17,-1$, so multiplicativity gives all four. The <two-isogeny formula> companion $E':y^2=x^3+68x$ has only positive candidate classes $\{1,2,17,34\}$; $(0,0)$ realizes $[68]=17$ and $(2,12)$ realizes $2$. Thus both images have order four, and the <two-isogeny rank formula> gives $2^r=4\cdot4/4=4$. The <rank of an elliptic curve> is therefore two, without any unresolved covering class.
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