 Rank-one elliptic curve with coefficients eight and minus seven

The two-isogeny formula gives the partner $E^{\prime }:y^2=x^3-16x^2+92x$. 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 $E^{\prime }$, $(x-8{)}^2+28>0$ 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 $2^r=2\cdot 4/4=2$, 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.