#15115: correct set of points at infinity for hyperelliptic curve
-------------------------------------+-------------------------------------
Reporter: mstreng | Owner:
Type: defect | Status: new
Priority: major | Milestone: sage-5.12
Component: geometry | Resolution:
Keywords: hyperelliptic | Merged in:
curve points at infinity | Reviewers:
Authors: Marco Streng | Work issues: doctest the rest of
Report Upstream: N/A | the sage library, check how
Branch: | documentation looks, add example
Dependencies: #15108 | from ticket description to patch
| Commit:
| Stopgaps:
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Changes (by mstreng):
* dependencies: => #15108
Old description:
> Here are two isomorphic hyperelliptic curves:
> {{{
> sage: HyperellipticCurve(x^6+x-1).points()
> [(0 : 1 : 0),
> (1 : 6 : 1),
> (1 : 1 : 1),
> (2 : 4 : 1),
> (2 : 3 : 1),
> (4 : 5 : 1),
> (4 : 2 : 1)]
> sage: HyperellipticCurve(-x^6+x^5+1).points()
> [(0 : 1 : 0),
> (0 : 6 : 1),
> (0 : 1 : 1),
> (1 : 6 : 1),
> (1 : 1 : 1),
> (2 : 5 : 1),
> (2 : 2 : 1),
> (4 : 4 : 1),
> (4 : 3 : 1)]
> }}}
> The isomorphism is {{{(x,y) |-> (1/x,y/x^3)}}}. They don't have the same
> number of points because of singularities at infinity. But by
> "hyperelliptic curve", one really means the desingularized curve. So the
> function "points" should return points on the desingularized curve.
>
> See also #11800 and #11980
>
> Apply:
> * [attachment:15115.patch]
New description:
Here are two isomorphic hyperelliptic curves:
{{{
sage: HyperellipticCurve(x^6+x-1).points()
[(0 : 1 : 0),
(1 : 6 : 1),
(1 : 1 : 1),
(2 : 4 : 1),
(2 : 3 : 1),
(4 : 5 : 1),
(4 : 2 : 1)]
sage: HyperellipticCurve(-x^6+x^5+1).points()
[(0 : 1 : 0),
(0 : 6 : 1),
(0 : 1 : 1),
(1 : 6 : 1),
(1 : 1 : 1),
(2 : 5 : 1),
(2 : 2 : 1),
(4 : 4 : 1),
(4 : 3 : 1)]
}}}
The isomorphism is {{{(x,y) |-> (1/x,y/x^3)}}}. They don't have the same
number of points because of singularities at infinity. But by
"hyperelliptic curve", one really means the desingularized curve. So the
function "points" should return points on the desingularized curve.
See also #11800 and #11980
Apply:
* [attachment:15115.patch]
Based on top of #15108 (so may depend on that).
--
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Ticket URL: <http://trac.sagemath.org/ticket/15115#comment:2>
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