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6.4. Class polynomials for genus 2 183and Streng [252] for details; and to the paper of Freeman, Stevenhagen and Streng [102] for theordinary case, and that of Hitt O’Connor et al. [134] for the p-rank one case.We have seen that the CM order is preserved by the Galois action, so the correspondingIgusa class polynomials computed over all isomorphism classes of complex abelian varieties withcomplex multiplication by a given order will have rational coefficients. Moreover, invertible idealsare sent onto invertible ideals, so that we can restrict ourselves to such ideals, even though thefollowing algorithms can be naturally extended to orbits of non-invertible proper ideals, and getfactors of the Igusa class polynomials.The first algorithm we propose is thus a straightforward generalization of the constructionof the Hilbert class polynomial in dimension 1: one first enumerates all isomorphism classes ofp.p.a.v. for a given order O and given by an invertible ideal and then computes the correspondinginvariants analytically with sufficient precision. The Igusa class polynomials, which have rationalcoefficients, can then be recognized from their approximations. Precise description and analysisof such an algorithm have been done by Streng in his Ph.D. thesis [252] in the case of maximalorders and readily extend to the case of non-maximal orders. The enumeration of isomorphismclasses is described in Algorithm 6.2 which corresponds to the algorithm proposed by Streng inthe case of maximal orders [252, Algorithm II.3.1]. Its correctness is a consequence of the resultspresented in Section 6.3. A high-level description of the algorithm is given in Algorithm 6.3Algorithm 6.2: Computation of representatives of the isomorphism classes of p.p.a.v. withCM by an order O in a primitive CM quartic field and given by invertible idealsInput: A primitive quartic CM field K and an order OOutput: A triple (Φ, a, ξ) for each isomorphism class of p.p.a.v. with CM by O1 Compute a complete set T of representatives of the CM types Φ on K2 Compute a complete set U of representatives of O ∗ K 0/ N K/K0 (O ∗ )3 Compute a complete set I of representatives of the Picard group of O4 Initialize an empty list L = []5 foreach Ideal a ∈ I do6 if a ∗ a −1 is principal and generated by an element ξ ∈ K such that ξ 2 ∈ K 0 and ξ istotally negative then7 Append (a, ξ) to A8 Initialize an empty list M = []9 foreach Pair (a, ξ) ∈ L and unit u ∈ U do10 Append (Φ, a, uξ) to M where Φ is the unique CM type Φ such that I(φuξ) > 011 return The triples (Φ, a, uξ) ∈ M such that Φ ∈ Twhich is nothing but an extension of the algorithm of Streng [252, Algorithm II.11.1].A second version of this algorithm using the explicit description of the main theorem ofcomplex multiplication over the reflex field was developed by Streng [252, Chapter III]. The ideais that, if σ ∈ Aut(C/K r ), then it does not modify the type Φ and acts as multiplication by thereflex norm N Φ r(s −1 ) of the corresponding idèle s. This action can be explicitly described asmultiplication by N Φ r(a −1 ) where a is an ideal of O K r. Therefore, it is enough to find a triple(Φ, a, ξ) and then to compute the action of Pic(O K r) on it to build irreducible components of theIgusa class polynomials over K r corresponding to a single orbit under Aut(C/K r ). In fact, it canbe shown that any K r 0-conjugate of an Igusa invariant is also a K r -conjugate [252, Corollary I.9.3].Therefore, these irreducible components of the Igusa class polynomials have coefficients in K r 0.One should then recognize the coefficients in K r 0.