= Solution
Write $x=c/d$ in lowest terms and put $H=\max\{|c|,|d|\}\ge1$. For $P\ne O,(0,0)$,
$$
x(\phi(P))=\frac{c^2+acd+bd^2}{cd}.
$$
Before reduction to lowest terms, both numerator and denominator are bounded in magnitude by $(1+|a|+|b|)H^2$. Cancellation can only decrease the height. Hence one may choose
$$
\boxed{k=\log(1+|a|+|b|).}
$$
The duplication formula for $C$ is
$$
x(2P)=\frac{(x^2-b)^2}{4x(x^2+ax+b)}
=\frac{(c^2-bd^2)^2}{4cd(c^2+acd+bd^2)}.
$$
The numerator has magnitude at most $(1+|b|)^2H^4$, and the denominator at most $4(1+|a|+|b|)H^4$. Thus
$$
\boxed{\ell=\log\max\{(1+|b|)^2,\ 4(1+|a|+|b|)\}}
$$
satisfies the second bound. At the identity or any point whose relevant image is the identity, the image height is zero and the same nonnegative constants work directly. Thus the formulas cover every rational point, including the exceptional points of the displayed coordinate expressions.
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