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152 Chapter 5. Complex multiplication and elliptic curvesTheorem 5.3.12 (Main theorem of complex multiplication [245, Theorem II.8.2], [160, Theorem10.2.3]). Let K be an imaginary quadratic field and O an order in K. Let E be an ellipticcurve defined over C with complex multiplication by O. Let σ ∈ Aut(C) be an automorphism ofC and s ∈ A ∗ K an idèle of K such that (s, K) K = σ. Fix a proper ideal a of O and a complexabanalytic isomorphismθ : C/a → E(C) .Then there exists a unique complex analytic isomorphismsuch that the following diagram commutes:θ ′ : C/s −1 a → E σ (C)K/as −1K/s −1 aθ θ ′E(C)σE σ (C)5.3.4 Computing the Hilbert class polynomialTo conclude this section, we give the outline of the computation of the Hilbert class polynomialusing the so-called complex analytic method in Algorithm 5.2.Algorithm 5.2: Computation of the Hilbert class polynomialInput: A negative discriminant ∆ ≡ 0, 1 (mod 4)Output: The Hilbert class polynomial H ∆ (X) of the order of discriminant ∆1 Compute a basis of the order O of discriminant ∆2 Compute the class group Pic(O) of O3 foreach a ∈ Pic(O) do4 Compute j(a) with enough precision5 Construct H(X) ∈ Z[X] from the complex approximations of its roots6 return H(X)Computation of the class group in Line 2 can be done working with binary quadratic forms.Computation of the j-invariant in Line 4 is made considering it as a modular function andusing its q-expansion [245, Proposition I.7.4], [160, Section 4.2].The complex analytic method goes back to the work of Atkin and Morain [9]. A precisedescription and analysis of a sophisticated version of Algorithm 5.2 can be found in the work ofEnge [85]. It is shown that, under the heuristic assumption that the correctness of the algorithmdoes not depend on rounding errors, the Hilbert class polynomial can be computed in O(h 2+ɛ )operations for any ɛ > 0 [85, Corollary 1.3], which is asymptotically optimal.Finally, it should be noted that other methods have been proposed to compute the Hilbertclass polynomial:• a p-adic method, avoiding numerical instability issues and also asymptotically optimal,first described in the work of Couveignes and Hencocq [60] and analyzed in a paper ofBröker [28];