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

Reducing the complexity for class group computations using small defining polynomials