Past exam of the analytic number theory and elliptic curves course of the University of Oxford / 2008 / B / 7 / ii / Solution

For rank-one two-isogeny descent with parameters three and minus three, use the standard two-isogeny rank formula. For $C:y^2=x(x^2+ax+b)$ and its dual curve $D:y^2=x(x^2-2ax+a^2-4b)$, let ${\alpha }_C,{\alpha }_D$ be the abscissa square-class maps, with their usual values at $(0,0)$ and the identity. If $r_C$ is the Mordell–Weil rank, thenFor the present curve $a=3,b=-3$, its possible classes are $1,-1,3,-3$, and the identity and $(0,0)$ supply $1,-3$. Exclude the class three as follows. A point in that class has $x=3(U/V{)}^2$, with coprime integers $U,V$, and substitution in the curve gives the necessary quartic equationwith integer $N$. Modulo three this forces $3\mid V,N$. Coprimality then makes $U$ a unit modulo three, and modulo nine the right side is $3U^4\equiv 3$, while $N^2\equiv 0$, a contradiction. The class minus one is excluded as well, since multiplying it by the known class minus three would give three. Therefore the first image has size two.

The dual curve is $D:y^2=x(x^2-6x+21)$. Its quadratic factor is $(x-3{)}^2+12>0$ on the real line, so every nonzero rational abscissa is positive. By (i) its image is contained in $\{1,3,7,21\}$. The point $(0,0)$ supplies 21 and the point $(3,6)$ supplies three. As the square-class map is a homomorphism, their product supplies seven. All four occur. Hence the descent formula gives $2^{r_C+2}=2\cdot 4=8$, so