= Solution
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.
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