This problem comes from Koblitz's book Introduction to Elliptic Curves and Modular Forms page 41 problem 2 and it says
Let $$f_N(z) = N \Pi(\wp(z)-\wp(u))$$
Where the product is taken over nonzero $u \in \mathbb{C} / L$ such that $Nu \in L$, with only one $u$ taken from the pair of $u$ and $-u$. So if $N$ is odd, we should have that $f_N(z)$ as a polynomial in $\wp(z)$ should have degree $\frac{N^2-1}{2}$.
My task is to try and find $f_3(z)$ in terms of $\wp(z)$, for the elliptic curve given in Weirestraus form $y^2 = 4x^3-g_2x-g_3$.
I know that it should be a degree $4$ polynomial with leading term $3\wp(z)^3$, and that we could set this up as a product in terms of the generators of the lattice $\omega_1, \omega_2$, but I can't seem to get this into a rational function in just terms of $\wp(z)$ and the constants $g_2$ and $g_3$.
I don't need a full solution, just maybe a nudge in the right direction, I'm trying to self teach myself the topics in this book. Thanks!
Let $u_i$ be representatives of order $3$ points modulo $\pm$. Note that $(\wp(u_i),\wp'(u_i))$ lies on the elliptic curve $y^2 = 4x^3-g_2x-g_3$.
For $(x,y)$ on the curve, the $x$-coordinate of its double is: $$\frac{g_2^2+8 g_2 x^2+32 g_3 x+16 x^4}{16 y^2}$$
If $(x,y)$ is a point of order $3$, then the above expression equals $x$, which means $$-g_2^2 -48 g_3 x -24 g_2 x^2 + 48x^4 = 0$$
$\wp(u_i)$ satisfies the above equation, and they are distinct, hence $$\prod_{i=1}^4 (z-\wp(u_i))= z^4 -\frac{g_2}{2} z^2 - g_3 z - \frac{g_2^2}{48}$$