= Rank-one elliptic curve with coefficients five and minus five
{title2=$E:y^2=x^3+5x^2-5x,\quad r=1$}
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 $E':Y^2=X^3-10X^2+45X$. Positivity and <prime> support allow only $1,3,5,15$. Class five is realized by $(0,0)$, whereas class three would require $N^2=3U^4-10U^2V^2+15V^4$. 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 $2^r=4\cdot2/4=2$.
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