Some remarks on the construction of class polynomials

Class invariants are singular values of modular functions which generate the class fields of imaginary quadratic number fields. Their minimal polynomials, called class polynomials, are uniquely determined by a discriminant $-D<0$ and are used in many applications, including the generation of elliptic curves. In all these applications, it is desirable that the size of the polynomials is as small as possible. Among all known class polynomials, Weber polynomials constructed with discriminants $-D \equiv 1$ (mod $8$) have the smallest height and require the least precision for their construction. In this paper, we will show that this fact does not necessarily lead to the most efficient computations, since the congruences modulo $8$ of the discriminants affect the degrees of the polynomials.

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