nt.number theory - Values of the j-function

One crude but effective method is to compute all the h conjugates a_j numerically to high precision, from which you can find the polynomial Π(x-a_j) they are the roots of using the fact that it has integral coefficients, (h=class number, and the values a_j 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.