Magma code

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Invariants and reconstruction of genus 3 hyperelliptic curves

We provide new functionalities for the package created by Reynald Lercier & Christophe Ritzenthaler for the genus 3 hyperelliptic curves. Now the curves defined over a field of characteristic 3 or 7 are supported. It has been implemented in Magma (v.2.20). You can download the .tgz here.

Short description : It provides (in characteristic not equal to 2 or 5) the computation of a set of generators for the ring of invariants (the Shioda invariants or SL2-invariants) and construct a hyperelliptic curve from given invariants. It also gives the geometric automorphism group of the curve (as an abstract group). Note that this works also for family of curves. The construction takes place over the field of moduli when there is no obstruction and over a quadratic field otherwise. Over a finite field (of characteristic not equal to 2 or 5), one can also compute the twists.