Birational Transformations question

Question

so I'm wondering, is there a birational transformation one can make to the equation $Y^2 = X^m + f_{m-1}X^{m-1} + ... + f_0$, where all $f_i \in \mathbb{Q}$ so it is of the form $Y^2 = X^m + g_{m-1}X^{m-1} + ... + g_0$, where all $g_i \in \mathbb{Z}$?

any help greatly appreciated, thanks!

Answer 1

Sure you can you do the following:

1. Find the lowest common multiple of the denominators.
2. Multiply the result by a constant so that it becomes a power of $6$.
3. Multiply the result in the equation.
4. Linear transformation.

E.g. $$y^2=x^3+\frac{2}{9}x^2+\frac{1}{8}x+2$$

Step 1: $LCM = 72=2^3\cdot3^2$.
Step 2: Let $N=2^3\cdot3^4\cdot LCM=46656$.
Step 3/4: \begin{align*} 46656y^2&=46656x^3+10368x^2+5832x+93312\\ (8\cdot27y)^2&=(4\cdot9x)^3+8(4\cdot9x)^2+162(4\cdot9x)+93312 \end{align*}