I encountered a question about showing that the curve $$ y^2 = x^4 + a_3 x^3 + a_2x^2 + a_1x + a_0, \qquad\qquad(1) $$ where $a_i \in \mathbb{Q}$, can be birationally transformed over $\mathbb{Q}$ to a curve of the form $$ y^2 = x^3 + Ax + B \qquad \qquad (2) $$ where $A, B \in \mathbb{Z}$. The hint was to write equation (1) in the form $y^2 = g(x)^2 + h(x)$, where $g(x)$ is quadratic and $h(x)$ is linear, and then transforming using $T = y+g(x), S=x(y+g(x))$.
This was fine. I was then asked to execute this procedure on the curve $y^2 = 2x^2 + 9$, by first converting it into the form of (1) (i.e. making it monic). After some work I was able to transform it into $$ y^2 = x^3 + 5832 x, $$ which, as you can see, is now in the form of equation (2) (perhaps someone can check this is correct).
The final part was to show that $y^2 = 2x^4 + 7$ can "also be birationally transformed over $\mathbb{Q}$ to the same form". It was tough to see how to do this at first. But I used the transformation that replaces $x$ by $x+1$, giving the constant term $9$ in the polynomial, and tried to follow the same method as above. However, it quickly becomes very messy. Is this the right way to go about doing it? Or is there some slick deduction from the first part I am missing?
