#5976: [with patch; needs work] Add an Elliptic Curve Isogeny object
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Reporter: shumow | Owner: shumow
Type: enhancement | Status: assigned
Priority: major | Milestone: sage-4.0
Component: number theory | Keywords: Elliptic Curves
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Comment(by kohel):
Just one additional comment on John's message: we could add "model" as
an option to specify the desired model of the codomain: "minimal_model",
"short_weierstrass", etc.; in characteristic 2 there are some special
Weierstrass models used in crypto.
There are numerous other models which are becoming trendy in crypto
applications: http://www.hyperelliptic.org/EFD/
But not all of them are isogeny invariant, even over a finite field.
E.g. the Edward's model has a rational 4-torsion subgroup, which
might change to a subgroup Z/2Z x Z/2Z under a 2-isogeny, but it is
stable under odd degree isogenies.
Although such special models are not yet implemented as classes in
Sage, it is good to keep them in mind when determining the optional
parameters one might want to feed into an isogeny. Obviously one
would also want to generalise the isomorphisms [u,r.s,t] of the
Weierstrass models (as well as the translation-by-P maps which are
isomorphisms of curves but not of elliptic curves, since they don't
fix the base point).
--David
P.S. I have a question:
If one starts with a minimal Weierstrass model (over QQ or a number
field), and applies a Velu isogeny phi (fixing a1, a2, a3) with pullback
scalar 1. How close or far from a minimal model is it -- e.g. is it
immediately p-minimal for all p coprime to the deg(phi)? What happens
at 2 and 3? John's argument shows that we can lose p-minimality since
the Velu isogeny with kernel E[p] is not [p] which has pullback scalar p,
so we must lose minimality at p. In short, how much or how little work
is needed to transform the codomain to a minimal model?
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/5976#comment:17>
Sage <http://sagemath.org/>
Sage - Open Source Mathematical Software: Building the Car Instead of
Reinventing the Wheel
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