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Given $\tau\in H$ (up-half plane) and $q=e^{2\pi i \tau}$, Weber polynomail is defined as

$$f(\tau)=q^{-\frac{1}{48}}\prod_{i=0}^{\infty}(1+q^{i-\frac{1}{2}}).$$

My question is: How can I compute a product of unlimited sequence? Anyway, I have to finish $f(\tau)$ in finite steps. Of course, the faster the better.

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  • $\begingroup$ $f(\tau)$ is not a Weber polynomial, but a Weber function. $\endgroup$
    – Alex M.
    Commented May 4, 2018 at 13:15

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An efficient construction of Weber polynomials, including a publicly available code, is given on page 340 and following of On the Efficient Generation of Elliptic Curves over Prime Fields (2003).

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  • $\begingroup$ Thanks a lot! I still have some small questions above your paper: (1) To compute $\Delta(\tau)$ on page 339, do you mean to approximate $\sum_{n\geq 1}...$ using Taylor method? (2) Does the defintion of $F(z)$ on page 340 say that $F(z)=1+\sum_{n\geq 1}...$ by cancelling the $z$ and the exponents $24$ and $1/24$? (3) What are $G$ and $I$ mentioned on page 341? $\endgroup$
    – Licheng Wang
    Commented May 5, 2018 at 2:07

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