state-of-the-art bit complexity of the determinant
Storjohann designs a Las-Vegas algorithm with $\tilde O(n^\omega M)$ bit operations
http://dx.doi.org/10.1016/j.jco.2005.04.002

Prior to this, Kaltofen and Villard gave improved algorithms, see 
http://lara.inist.fr/bitstream/handle/2332/850/LIP-RR2003-36.pdf%3Fsequence%3D1