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184 Chapter 6. Complex multiplication in higher generaAlgorithm 6.3: Computation of the Igusa class polynomials of a non-maximal orderInput: An order O in a CM quartic number field KOutput: The Igusa class polynomials H 1 (X), H 2 (X) and H 3 (X) corresponding to thePicard group of O1 Compute a set M of representatives of the isomorphism classes of p.p.a.v. with CM by Ogiven by invertible ideals2 foreach Triple (Φ, a, ξ) ∈ M do3 Compute a symplectic basis of a with respect to the polarization defined by ξ4 Reduce the corresponding period matrix Ω into a fundamental domain5 Evaluate the Igusa invariants i 1 , i 2 and i 3 with sufficient precision through analyticevaluation of theta constants on Ω6 Reconstruct the polynomials H 1 (X), H 2 (X) and H 3 (X) with rational coefficients fromtheir respective roots7 return H 1 (X), H 2 (X), H 3 (X)This method can as well be extended to the case of non-maximal orders. Indeed, we know thatσ ∈ Aut(C/K r ) also acts as multiplication by the idèle N Φ r(s −1 ) for general orders. To describethat action explicitly as multiplication by an invertible ideal, it is sufficient that N Φ r(s −1 ) ≡ 1mod f, which can always be achieved up to multiplication by a principal idèle. Equivalently, theaction of an ideal a of O K r is given by multiplication N Φ r(a −1 ) ∩ O provided that N Φ r(a −1 ) isco-prime to f. This can also always be achieved up to multiplication by a principal ideal.6.4.4 Going furtherThe previous description of the algorithms is very scarce and many details are omitted. The missingsteps for the principal case can be found in the Ph.D. thesis of Streng [252] and extend naturallyto the non-principal case. We now list a few of them as well as some practical optimizations.A first concrete complication in comparison with the genus 1 case is that Igusa class polynomialsare no more integral, but only rational. Moreover, computing bounds on the value of thedenominator is a hard problem. It was recently achieved by Goren and Lauter [119, 120] eventhogh their result is not very sharp.Furthermore, once the isomorphisms classes and the corresponding period matrices havebeen computed, they should be reduced, i.e. moved to a fundamental domain, to obtain fasterconvergence as is done for genus 1 [252, II.5].Then, the theta constants should be computed using Borchardt sequences as proposed byDupont [75]. This led to some record computations made by Dupont, Enge and Thomé [76].The complex analytic method is not the only one which extends from genus 1. Eisenträgerand Lauter developed a CRT method [80], implemented by Freeman [103] and later improved byBröker, Gruenewald and Lauter [29] and Lauter and Robert [166]. As far as the p-adic methodsare concerned, Gaudry et al. [112] developed a 2-adic one and Carls, Kohel and Lubicz [43] a3-adic one.To conclude, it should be mentioned that it is possible to define smaller class invariants thanthe Igusa invariants as was already the case in dimension 1. Such an approach, based on thehigher-dimensional Shimura reciprocity, is being conducted by Streng [254, 255, 253]. Relatedideas can also be found in the Ph.D. thesis of Uzunkol [269].