#7262: Have multiplcation_by_m() return an EllipticCurveIsogeny object
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Reporter: wuthrich | Owner: davidloeffler
Type: defect | Status: new
Priority: minor | Milestone: sage-4.2
Component: elliptic curves | Keywords: elliptic curves, isogeny,
Work_issues: | Author:
Reviewer: | Merged:
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Comment(by wuthrich):
moving an email discussion forward, John said :
> So, do we want to allow any (separable) isogeny to be created from the
> full x-coordinate rational map (or from its numerator and
> denominator)? That soulds like a good idea _provided_ that (1) we can
> do it, more cheaply that discarding the numerator and calling existing
> code, and also perhaps (2) we have a way of checking the validity.
I would send both coordinates to the constructor. Your code of
constructing them is faster than what the generic isogeny-code will do.
> As a first step, suppose we only allow this when the codomain is also
> specified (as we can do in this case). Then checking validity is
> quite easy (you get the y-coordinate by differentiating and scaling
> appropriately and then plug into the equation of the codomain -- I can
> write this out properly if you do not follow me.)
Yes, that is what I will do. I do not think that we will need this option
of specifying an isogeny when the codomain is not known. One could always
send the denominator as a kernel_polynomial which will create the
codomain.
Of course there is an obvious extension that should be added here. Complex
multiplication, like multiplication_by_n(2*i+3). Also we could make
automorphisms into a group and create a endomorphism ring... oh but I am
dreaming some steps ahead of what I will do about this ticket.
Chris.
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Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/7262#comment:1>
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