= Rank-one elliptic curve with coefficients eight and minus seven
{title2=$E:y^2=x^3+8x^2-7x$}
The <two-isogeny formula> gives the partner $E':y^2=x^3-16x^2+92x$. The <two-torsion square-class homomorphism> image for $E$ is $\{1,-7\}$: <prime> support leaves only $\{1,-1,7,-7\}$, the <modulo-sixteen obstruction for the negative unit descent class> excludes $-1$, and the subgroup property excludes $7$. For $E'$, $(x-8)^2+28>0$ forces nonnegative $x$, so the image is contained in $\{1,2,23,46\}$. The two-torsion point realizes $23$ and the <rational point> $(18,48)$ realizes $2$, giving all four classes. The <two-isogeny rank formula> yields $2^r=2\cdot4/4=2$, hence rank one. The <dual isogeny> sends $(18,48)$ to $(16/9,116/27)$; this statement does not assert that the resulting point is a generator of the full free part.
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