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We describe a new algorithm for computing the ideal class group, theregulator and a system of fundamental units in number fields under thegeneralized Riemann hypothesis. We use sieving techniques adapted from thenumber field sieve algorithm to derive relations between elements of the idealclass group, and $p$-adic approximations to manage the loss of precision duringthe computation of units. This new algorithm is particularily efficient fornumber fields of small degree for which a speed-up of an order of magnitude isachieved with respect to the standard methods.

New techniques for computing the ideal class group and a system of fundamental units in number fields