Suppose we have an elliptic curve of the form:
$y^{2}=x^{3}-f\left(z_{1},z_{2}\right)x-g\left(z_{1},z_{2}\right)$
This describes a Calabi-Yau threefold as an elliptic fibration over $\mathbb{P}^{1}\times\mathbb{P}^{1}$, where $\left(z_{1},z_{2}\right)$ are the coordinates of the $\mathbb{P}^{1}\times\mathbb{P}^{1}$; $f$ is a homogenous polynomial of degree 8 in each of the $z_i$; and $g$ is a homogenous polynomial of degree 12 in each of the $z_i$. The $j$-invariant should be:
$j=\frac{4f^{3}}{4f^{3}-27g^{2}}$
Now suppose that this $j$-invariant is also expressed in terms of $t\left(z_{1},z_{2}\right)$. Specifically, I am looking at the Index 36 $j$-invariants on page 5 here: http://mysite.science.uottawa.ca/asebbar/publi/mcse.pdf.
I want to find a substitution of the form:
$t=\frac{P\left(z_{1},z_{2}\right)}{Q\left(z_{1},z_{2}\right)}$
Where $P$ and $Q$ are polynomials of some appropriate degree, such that we reproduce the form the $j\left(f,g\right)$ above. Is there any general procedure for doing this? Or is it just frustrating guesswork?
Many thanks!
