RAMANUJAN INVARIANTS FOR DISCRIMINANTS CONGRUENT TO 5 (mod 24)

Abstract

In this paper we compute the minimal polynomials of Ramanujan values [Formula: see text] for discriminants D ≡ 5 ( mod 24). Our method is based on Shimura Reciprocity Law as which was made computationally explicit by Gee and Stevenhagen in [Generating class fields using Shimura reciprocity, in Algorithmic Number Theory, Lecture Notes in Computer Science, Vol. 1423 (Springer, Berlin, 1998), pp. 441–453; MR MR1726092 (2000m:11112)]. However, since these Ramanujan values are not class invariants, we present a modification of the method used in [Generating class fields using Shimura reciprocity, in Algorithmic Number Theory, Lecture Notes in Computer Science, Vol. 1423 (Springer, Berlin, 1998), pp. 441–453; MR MR1726092 (2000m:11112)] which can be applied on modular functions that do not necessarily yield class invariants.