Abstract: Matzat and Malle proved that the Matthieu group M24 is Galois over Q. They build a non rigid covering and prove the existence of a rational point in an adequate Hurwitz space. We give here an explicit extension with these properties. We deduce also the existence of a regular extension of K(T) with Galois group M23 for any field K such that x2+y2+z2=0 has a non trivial solution. The objects computed are too large for symbolic calculation. To obtain our results, we use numerical methods and then we deduce the algebraic values. This illustrates the efficiency of the techniques developped by Couveignes and Granboulan. [1, 2].