An admissible change of <Weierstrass equation of an elliptic curve> replaces the $x$-coordinate by a degree-one rational function, so the two logarithmic heights differ by $O(1)$. Dividing this bounded difference by $4^r$ in the defining limit shows that the canonical height is unchanged. It is intrinsic to the elliptic curve and the chosen point.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 125 / 4 / c / i / Solution

 Solution (source code)