Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 125 / 4 / b / Solution

Foruse two-isogeny descent throughThe square-class maps send a nonexceptional point to the class of its $x$-coordinate. On $E$, possible classes are $\pm 1,\pm 7$; the defining quartics and positivity exclude the negative classes, while $(1,3)$ and $(7,21)$ realize $1$ and $7$. Thus the image has order two. On $E^{\prime }$, the possible classes are $\pm 1,\pm 3$; the classes $1$ and $-3$ occur, while the quartics for $-1$ and $3$ have no primitive solution modulo $16$. Hence this image also has order two. The two-isogeny descent formulatherefore gives $r=0$.

The displayed curve has good reduction at $5$ and $11$, where direct counting givesReduction injects rational torsion of order prime to these characteristics, so its order divides $\gcd (6,18)=6$. We already have the six distinct pointsSince the rank is zero, these are all the rational points and $E(\mathbb{Q})\cong \mathbb{Z}/6\mathbb{Z}$.