I‑AIMS Institute for Artificial Intelligence
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A Maple package for hyperelliptic curves

Tanush Shaska, S. Zheng

I-AIMS paper 2005-4 · Published in (2005), 399–408 · 2005

We implement a Maple package that addresses questions of the arithmetic of hyperelliptic curves. Our package has as main functions the following: determining the automorphism group of a genus g hyperelliptic curve 𝒳_g defined over ℂ, computing the isomorphism class [𝒳_g] for |Aut(𝒳_g)| > 2, determining the field of moduli and the field of definition, and giving an explicit equation of the curve over ℚ when such equation exists. We can accomplish such tasks for many classes of hyperelliptic curves. For genus g = 2 we can answer these questions completely. Given a genus 2 curve C defined over ℂ, this package accomplishes the following: computes the moduli point 𝔭(C) ∈ ℳ₂, determines the automorphism group of C, determines if a rational model of C over ℚ exists, computes such rational model when it exists, for n = 2, 3, 5 determines if the Jacobian of C is (n,n)-split, determines if C has small degree elliptic subcovers and computes such subcovers. To accomplish such tasks we have implemented many other functions such as Igusa invariants of binary sextics, absolute invariants, etc. We can do most of the above for genus 3 curves with automorphism group of order > 4. Conditions in terms of classical invariants of binary forms are implemented which check if a curve has reduced automorphism groups A₄, S₄, A₅.

MSC classes: Primary 65D18; Secondary 14Q05, 14H52, 65Y15, 68W30

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References

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  15. T. Shaska, Curves of genus 2 with (n, n)-decomposable Jacobians, J. Symbolic Comput. 31 (2001), no. 5, 603–617.

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