#11800: Problem with points at infinity in hyperelliptic curves
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Reporter: gaudry | Owner: AlexGhitza
Type: defect | Status: needs_work
Priority: minor | Milestone: sage-4.8
Component: algebraic geometry | Keywords: ecc2011, sd35,
hyperelliptic curve, conic
Work_issues: | Upstream: N/A
Reviewer: Marco Streng | Author: David Eklund
Merged: | Dependencies: #11930
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Changes (by mstreng):
* status: needs_review => needs_work
* reviewer: => Marco Streng
Comment:
Points at infinity are counted incorrectly for degree 2 by this patch.
{{{
sage: C = Conic(GF(7), [1, 0, 0, -1, 0, 1])
sage: R.<x> = GF(7)[]
sage: H = HyperellipticCurve(x^2+1)
sage: C
Projective Conic Curve over Finite Field of size 7 defined by x^2 - y^2 +
z^2
sage: H
Hyperelliptic Curve over Finite Field of size 7 defined by y^2 = x^2 + 1
sage: C.is_smooth()
True
sage: H.points()
[(0 : 6 : 1), (0 : 1 : 1), (1 : 4 : 1), (1 : 3 : 1), (6 : 4 : 1), (6 : 3 :
1)]
sage: H([1,1,0])
(1 : 1 : 0)
sage: H([1,-1,0])
(6 : 1 : 0)
}}}
Here C and H represent the same curve. It is a smooth conic over a finite
field of order 7, hence has 8 rational points, but only 6 are found.
In general, if H has degree 2, there are 0 or 2 rational points at
infinity (as in my previous comment). Since H has degree < 4, the plane
model equals the smooth model, so these points can be represented and
returned correctly in Sage.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/11800#comment:15>
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