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**Example text**

E. ℓ | [OK : Z[π]] and ℓ2 ∤ [OK : Z[π]]), k the element nℓ α can be replaced by an element of the form π ℓ−1 . This replacement is useful since by [11, Fact 10], determining whether an element k of the form π ℓ−1 is an endomorphism is equivalent to testing the field of definition of the ℓ-torsion. 4. Let A ⊂ B ⊂ C be abelian groups, with [C : A] finite. Let ℓ be a prime, and suppose ℓ divides [C : A] and ℓ2 does not divide [C : A]. Suppose there is some β ∈ B such that β ∈ A and ℓβ ∈ A. Then for any α ∈ C such that ℓα ∈ A, α ∈ B.

Flon and R. Oyono. Fast arithmetic on Jacobians of Picard curves. In Public Key Cryptography - PKC 2004, volume 2947 of LNCS, pages 55–68. Springer, 2004. 9. S. Flon, R. Oyono and C. Ritzenthaler. Rationality of the intersection points of a line with a plane quartic. In progress, 2007. 10. G. Frey and M. M¨ uller. Arithmetic of modular curves and applications. In Algorithmic Algebra and Number Theory, pages 11–48. Ed. , Springer-Verlag, Berlin, 1999. 11. S. D. Galbraith. Equations for modular curves.

1 we just have to replace f3 by f4 . Furthermore, if char(k) = 3, we let Y = y + h1 (x)/3 and we can assume that C is of the following form: Y 3 + h2 Y = f4 , with h2 and f4 as above. If in addition char(k) = 2, then we can assume that f4 has no x3 term. 4. Comments on implementation We deal in this part with an optimized implementation in the case of the existence of a rational flex. To make the algorithm more efficient, we use the following well known methods: (1) In order to reduce the number of field inversions, we use Montgomery’s trick to compute simultaneous inversions.