= Rank-zero elliptic curve with roots zero three and four
{title2=$\operatorname{rank}E(\mathbb Q)=0,\quad E:y^2=x(x-3)(x-4)$}
For this <elliptic curve>, the <two-isogeny formula> gives the partner $E':Y^2=X^3+14X^2+X$. The <prime-support bound in two-isogeny descent> and positivity of nonzero rational $x$ restrict the first <square class> image to $1,2,3,6$. The identity and the <rational points> $(2,2),(0,0),(6,6)$ realize these classes. The partner image is contained in $\{1,-1\}$, and the <negative unit obstruction for the fourteen-coefficient isogeny cover> excludes $-1$. The <two-isogeny rank formula> therefore gives $2^r=4\cdot1/4=1$.
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