According to R.A.Kazmi's dissertation ["Isogenies and Cryptography"](https://eprint.iacr.org/2010/550.pdf) (Page 22), given an isogeny degree $l$, the Müller modular polynomials are defined as 


$$G\_l(x,y)=\sum\_{r=0}^{l+1}\sum\_{k=0}^{v}a\_{r,k}x^ry^k\in\mathbb Z[x,y]$$


where $v=\frac{s(l-1)}{12}$, $s=\frac{12}{\gcd(l-1,12)}$, but the explanation of $a\_{r,k}$ is missing. How can I compute them?



How to compute the Müller modular polynomials?
The definition of the coefficients $a\_{r,k}$ is given by theorem III.17 on page 53 of ["Elliptic curves in cryptography"](https://www.cambridge.org/core/books/elliptic-curves-in-cryptography/16A2B60636EFA7EBCC3D5A5D01F28546) by I.F. Blake, G. Seroussi, N.P. Smart (Cambridge University Press, 1999) (also on [Google Books](https://books.google.com/books?id=0_vegzgyqGMC)).


With $v,s,l$ as in the question, define $f(\tau) = \left( \dfrac {\eta(\tau)} {\eta(l\tau)} \right)^{2s}$, where $\eta$ is [Dedekind's $\eta$ function](https://en.wikipedia.org/wiki/Dedekind_eta_function). If $j$ is the invariant function (see page 47 of the quoted book), then there exist numbers $a\_{r,k} \in \mathbb Z$ such that


$$\sum \_{r=0} ^{l+1} \sum \_{k=0} ^v a\_{r,k} f(\tau)^r j(lt)^k =0 \ .$$


Now just formally replace $f(\tau)$ with $x$ and $j(l\tau)$ with $y$ to get $G\_l(x,y)$.


Brief descriptions of the actual computation of these coefficients are given on page 54, and in ["Counting the number of points on elliptic curves over finite fields of characteristic greater than three"](http://www.shoup.net/papers/counting.pdf) by F. Lehmann, M. Maurer, V. Müller, V. Shoup.


