Rank-one elliptic curve with coefficients five and minus five ($E:y^2=x^3+5x^2-5x,r=1$)

The two-torsion square-class homomorphism image on $E$ is all of $\{1,-1,5,-5\}$, realized by the identity, $(0,0)$, $(-1,3)$ and multiplication of classes. Its two-isogeny formula companion is $E^{\prime }:Y^2=X^3-10X^2+45X$. Positivity and prime support allow only $1,3,5,15$. Class five is realized by $(0,0)$, whereas class three would require $N^2=3U^4-10U^2{V}^2+15V^4$. Reduction modulo three forces one of coprime $U,V$ to be divisible by three; either choice gives odd valuation one on the right, contradicting a square. Multiplication by class five also excludes fifteen. The two-isogeny rank formula gives $2^r=4\cdot 2/4=2$.