Let A 5 , respectively A 5 , be the only non-trivial central extension of A 5 by the cyclic group C 2 of order 2, respectively, the cyclic group C 4 of order 4. These groups can be described as groups of matrices. The group A 5 is isomorphic to Sl 2 (F 5 ) and the group A 5 to [x # GL 2 (F 5 ) : det(x)=\1]. In this paper, we give an algorithm to construct some A 5 and A 5 -extensions of Q, given as the splitting fields of polynomials of degree 24. Let us mention that a different approach to this problem has been developed by T. Crespo in [C1] and [C2]. In the last section, we compute the first coefficients of the L-functions of the irreducible 2-dimensional complex representations corresponding to these extensions. One can find tables for such L-functions on the Internet at http:wwwÂÂmath.u-bordeaux . fr t jehanneÂ.
The results that we describe here take place in the theory of the complex Galois two-dimensional representations (see [F1], Sect. 4). For odd 2-dimensional representations, Artin's conjecture has been proved for infinitely many cases (see Theorem 1.1) by K. Buzzard, M. Dickinson, N. Shepherd-Barron and R. Taylor ([B-D-SB-T]), and more recently by R. Taylor in [Ta2] (we say that a complex Galois representation \ is even (resp. odd) if det()(c)=1 (resp. &1), where c is the complex conjugation). Hence our algorithm gives an efficient way to compute many modular forms of weight one corresponding to icosahedral representations (see [D-S]). Finally, it can be used for computing bases for spaces of modular forms of weight one. Moreover, in [J-M], a joint work with M. Mu ller, we use these constructions for giving a new experimental evidence of Artin's conjecture, in an unproved case of odd 2-dimensional representations. Even representations are more mysterious; we intend in another work to use our constructions for experimental comparisons between L-functions of icosahedral representations with Maa; forms.