#6193: [with patch, needs review] implement elliptic logarithm
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Reporter: robertwb | Owner: was
Type: defect | Status: new
Priority: major | Milestone: sage-4.0.1
Component: number theory | Keywords:
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Comment(by cremona):
Replying to [comment:2 robertwb]:
> The code looks good after my first reading.
>
> * I assume by {{{on_egg}}} you're implying the non-identity component
of an elliptic curve over R?
That is right. Some people call this (the compact component in {{{R^2}}})
the "egg". Perhaps a comment should be included to explain this, but the
name has the advantage of being short.
>
> * Where does the terminology {{{ei}}} come from for the x-coordinates
of the 2-torsion? (I may just not be familiar with the notation, if so,
just let me know.)
>
I thought it was standard to call the real roots e1, e2, e3 (i.e. these
are the x-coords of the points of order 2). Less standard is the ordering
(for curves over R): when they are all real then either e1<e2<e3 or the
other way round; and when only one is real, it is e1 for some people and
e3 for others. Hence I do make this explicit.
> * What assurance is there that {{{extended_agm_iteration}}} will
terminate in the presence of numerical noise? (I suppose if delta is
around machine epsilon, then (1+delta).sqrt() should be identically 1. Is
that enough?
>
That does worry me. I am hopeless at numerical analysis; I put this
simple test in while testing and it seemed to work fine; otherwise we
should be testing that delta is small enough that 1+delta is exactly 1
within the current precision. (Note that the way this is coded it is
already using relative rather than absolute precision, which is good).
> * It would be good to have an example demonstrating that the elliptic
log is actually the inverse of the standard Weierstrass isomorphism (at
least using Pari's version so far)
Of course; and that is listed in the things I have not done yet.
>
> I am still building a 4.0 so I haven't actually applied/tested it, but
will when that's done building.
>
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/6193#comment:3>
Sage <http://sagemath.org/>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
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