Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 125 / 5 / b / Solution

For $E:y^2=x^3-x^2+25x$, the addition formulas giveIn particular, $[2]Q=T$ and $Q$ has order four, while the nonintegral coordinate of $[2]P=(144/25,-2172/125)$ and part (a) show that $P$ has infinite order.

For two-isogeny descent, writeThe rational point $(0,0)$ is the kernel of a degree-two isogeny $E\to E^{\prime }$. The square-class maps send an affine point with $x\ne 0$ to $[x]\in {\mathbb{Q}}^{\ast }/{\mathbb{Q}}^{\ast 2}$, send $O$ to $1$, and send $(0,0)$ to $[b]$ or $[a^2-4b]$ on the two curves. Their images determine the rank through

HereOn $E$, the possible square classes divide $25$; real solubility excludes the negative classes, while $P$ and $Q$ exhibit $1$ and $5$. Thus $\alpha (E(\mathbb{Q}))=\{1,5\}$. On $E^{\prime }$, the pointsexhibit the classes $-11,-1,11$, along with $1$. The remaining candidate classes are divisible by $3$. Their homogeneous spaceshave no primitive solution modulo $9$: reduction modulo $3$ first forces $3\mid UV$, and then the equation is congruent to $3$ or $6$ modulo $9$. HenceThe rank formula gives $2^r=2\cdot 4/4=2$, so $r=1$.

At the good primes $7$ and $19$, direct counts giveThe reduction of torsion points on an elliptic curve injects rational torsion into both groups, so its order divides $\gcd (12,16)=4$. Since $Q$ has order four, the torsion subgroup is $\mathbb{Z}/4\mathbb{Z}$. Thereforeso $t=1$, $d_1=4$, and $r=1$.