Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 125 / 5 / d / Solution

By part b, writeThe canonical height of an elliptic curve vanishes on torsion and satisfies $\hat{h}(mG+T)=m^2\hat{h}(G)$. If $\hat{h}(P)=\hat{h}(Q)\ne 0$, their nonzero integer coefficients therefore have equal squares, sofor some $T\in E_5[2]$.

Negation leaves the $x$-coordinate unchanged. Addition by the four 2-torsion points changes $x$ amongSubstitution in the three side formulas for ${\Delta }_P$, using $5y^2=x^3-x$, only changes signs and permutes the three values. Hence ${\Delta }_Q$ and ${\Delta }_P$ are the same right triangle up to reordering their sides.