Reducing the complexity for class group computations using small defining polynomials

In this paper, we describe an algorithm that efficiently collect relations in class groups of number fields defined by a small defining polynomial. This conditional improvement consists in testing directly the smoothness of principal ideals generated by small algebraic integers. This strategy leads to an algorithm for computing the class group whose complexity is possibly as low as $L_{\mathrm{\mid }{\mathrm{\Delta }}_{\mathbf{K}}\mathrm{\mid }}\left(\frac{1}{3}\right)$.