#18611: Further isogeny improvement
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Reporter: jdemeyer | Owner:
Type: enhancement | Status: new
Priority: major | Milestone: sage-6.8
Component: elliptic curves | Resolution:
Keywords: | Merged in:
Authors: Jeroen Demeyer | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
Dependencies: #18589 | Stopgaps:
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Description changed by jdemeyer:
Old description:
> To compute all `l`-isogenies, Sage uses a function `mult(f)` which
> computes
> {{{
> gcd( numerator(f(m)), psi )
> }}}
> where `m` is a rational function giving the multiplication-by-`m` map.
>
> Instead of the above computation, the inverse direction is actually
> easier to compute: given `g`, we want to find `f` such that
> {{{
> gcd( numerator(f(m)), psi ) = g
> }}}
> Using some theory, this is equivalent to
> {{{
> f(m) = 0 mod g
> }}}
> Since `f` must be irreducible, this is just the characteristic (=
> minimal) polynomial of `a/b mod g`.
New description:
To compute all `l`-isogenies, Sage uses a function `mult(f)` which
computes
{{{
gcd( numerator(f(m)), psi )
}}}
where `m` is a rational function giving the multiplication-by-`m` map.
Instead of the above computation, the inverse direction is actually easier
to compute: given `g`, we want to find `f` such that
{{{
gcd( numerator(f(m)), psi ) = g
}}}
Using some theory, this is equivalent to
{{{
f(m) = 0 mod g
}}}
Since `f` must be irreducible, this is just the characteristic (= minimal)
polynomial of `m mod g`.
--
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Ticket URL: <http://trac.sagemath.org/ticket/18611#comment:2>
Sage <http://www.sagemath.org>
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