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5.2. Elliptic curves over the complex numbers 145As every fractional ideal of O is proper and so invertible for its multiplier ring, we get thefollowing proposition.Proposition 5.2.26. Let K be an imaginary quadratic field and O an order in K.H(O) =∑O⊂O ′ ⊂Kh(O ′ ) .Furthermore, for the maximal order, the fractional ideals are automatically proper, whencethe following proposition.Proposition 5.2.27. Let K be an imaginary quadratic field and O K its ring of integers. ThenPic(O K ) = Cl(O K ) and h(O K ) = H(O K ).Finally, the Picard group of an order can be described as a subgroup of the full class group ofK. First, we can restrict ourselves to the classes of ideals co-prime to the conductor.Proposition 5.2.28 ([160, Theorem 8.1.4], [61, Lemma 7.18]). Let K be an imaginary quadraticfield and O the order of conductor f in K. Let a be a non-zero integral ideal of O co-prime to f.Then a is proper.We denote by Frac(O, f) (respectively PF(O, f)) the subgroups of Frac(O) generated by integralideals (respectively principal integral ideals) co-prime to f.Proposition 5.2.29 ([160, Theorem 8.1.5], [61, Proposition 7.19]). Let K be an imaginaryquadratic field and O the order of conductor f in K. ThenPic(O) ≃ Frac(O, f)/ PF(O, f) .These ideals can then be pulled back to the maximal order of K.Proposition 5.2.30 ([160, Theorem 8.1.6], [61, Proposition 7.22]). Let K be an imaginaryquadratic field and O the order of conductor f in K. ThenPic(O) ≃ Frac(O K , f)/ PF Z (O K , f) ,where PF Z (O K , f) is the subgroup of Frac(O K ) generated by principal ideals a of O K such thatthere exist α ∈ O K and a ∈ Z with a = αO K , α ≡ a (mod fO K ) and gcd(a, f) = 1.We can deduce from the last proposition the classical expression of the class number of O interms of the class number of K.Theorem 3.2.12 ([160, Theorem 8.1.7], [61, Theorem 7.24]). Let K be an imaginary quadraticfield, O K its ring of integers and O the order of conductor f in K. Denote by ∆ the discriminantof K. Then(where·p)is the Kronecker symbol.h(O) = fh(O K)[O ∗ K : O∗ ]∏p|f( ( ) ∆ 11 −p p),