Values of the j-function In general, how do you compute the algebraic values of the modular j-function at quadratic imaginary points? (In other words, how do you compute the algebraic values of singular moduli?)
For instance, the Mathematica website (http://mathworld.wolfram.com/j-Function.html) gives the standard nine integral examples that result when the class number $h_k=1$, but then it gives 18 examples for when the class number is 2 without any specific references. How does one compute these? More importantly, can you also do it for higher degree cases? Or even just find the defining degree-$h_k$ polynomial?
 A: In case the Gross-Zagier paper doesn't meet your needs, you can also refer to the following 


*

*Harold Baier Efficient computation
of singular moduli 

*Noriko Yui 
    On The Singular Values Of Weber
    Modular Functions
and of course David Cox's book Primes of the form $x^2 + Ny^2$, Section 3.12
The bad thing about the j-function is that it is a level 1 modular function so the coefficients of its defining polynomial are going to explode with increasing degree.  Its easier to compute the singular moduli using a modular function of some higher level (e.g. Weber func has level 48) as demonstrated in the papers mentioned above as well in Cox's book.
A: One crude  but effective method is to compute all the h conjugates aj numerically to high precision, from which you can find the polynomial Π(x-aj) they are the roots of using the fact that it has integral coefficients, (h=class number, and the values aj are the values of j at the imaginary quadratic integers with the same discriminant.)
Alternatively see the paper On singular moduli by Gross and Zagier, which gives an explicit expression for the values of j as products of many small algebraic integers. 
