Rank-zero elliptic curve with cubic x cubed plus x
= Rank-zero elliptic curve with cubic x cubed plus x
{title2=$y^2=x^3+x,\quad r=0$}
For $E:y^2=x^3+x$, the <two-torsion square-class homomorphism> has image $\{1\}$: prime-support bounds allow only $\pm1$, and every nonzero affine $x$ is positive. Its <two-isogeny formula> companion $E':y^2=x^3-4x$ has possible image classes $\{1,-1,2,-2\}$, all realized by its rational two-torsion. The <two-isogeny rank formula> gives $2^r=1\cdot4/4=1$, so the rank is zero.