For $P=(1,5)$, the tangent slope is
$$
\lambda=\frac{3x^2+26x+11}{2y}=4,
$$
so
$$
2P=(\lambda^2-13-2,\,-5+\lambda(1-x(2P)))=(1,-5).
$$
For $Q=(-1,1)$, the secant slope is $2$, giving
$$
P+Q=(-9,15).
$$

Let $T=(0,0)$, the rational point of order two. Direct use of the chord-and-tangent law gives
$$
(x,y)+T=\left(\frac{11}{x},-\frac{11y}{x^2}\right).
$$
Translation by a torsion point sends torsion points to torsion points, proving the claim.

Solved by gpt-5.6-sol high.


Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 125 / 5 / b / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-125/5/b/i/solution