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6.2. Class groups and units 171in [151, Section 6]. The computation of the set P of maximal ideals of O containing the conductoris done by factoring f = fÕ in Õ:fÕ = qn1 1 · · · qnr r ,and then computing intersections of the q i with O:P = {q i ∩ O} 1≤i≤r.The first quotient to compute is nothing but(Õp/fÕp) ∗ ≃ ∏(Õq/fÕq) ∗ ≃ ∏(Õq/q nqÕ q) ∗ .q|pÕIt is not necessary to compute completely the second quotient. Indeed if pO p ⊃ fO p ⊃ p m O p , computinggenerators of (O p /p m O p ) ∗ is enough to deduce the structure of (Õp/fÕp) ∗ /(O p /fO p ) ∗ [151,Section 8]. The integer m is easily deduced from the factorization of fÕ and pÕ [151, Lemma 7.4].It now remains to explain the computation of (O p /p m O p ) ∗ for any order O, for a maximalideal p and a positive integer m. The first step is to get back to the global ring O with thefollowing classical result.Lemma 6.2.8 ([151, Theorem 4.1.i]). Let K be a number field and O an order. Let p be amaximal ideal and a a p-primary ideal. Thenq|pÕO/a ≃ O p /aO p .Afterwards, the structure of the unit group in the residue ring is given by the following lemma.Lemma 6.2.9 ([151, Lemma 4.3]). Let K be a number field and O an order. Let p be a maximalideal. Then(O/p m ) ∗ ≃ (O/p) ∗ × (1 + p)/(1 + p m ) .The computation of a generator of the finite field O/p is classical. It can then be lifted backto O/p m using Hensel’s lemma. The computation of (1 + p)/(1 + p m ) is done using a binarydecomposition and the following lemma.Lemma 6.2.10 ([151, Lemma 4.4]). Let K be a number field and O an order. Let a and b betwo ideals of O such that a ⊃ b ⊃ a 2 . Then the map ψ : (1 + a)/(1 + b) → a/b, [1 + γ] ↦→ [γ] isa group isomorphism.To summarize the above discussion, we finally recall the complete algorithm of Klüners andPauli [151, Algorithm 8.1] in Algorithm 6.1.The unit group is computed in a similar way using the fact that it is the kernel of the leftpart of the exact sequence1 → O ∗ → Õ∗ → ⊕ Õp/O ∗ p ∗ .pIt is then realized as the kernel of the map Õ∗ → ⊕ p Õ∗ p/O ∗ p.6.2.3 Multiplication of fractional ideals by finite idèlesThe main theorems of complex multiplication are expressed using the action of finite idèles onlattices which we describe in this subsection.The finite adèles and idèles are defined as follows.