#19122: cardinality_exhaustive incorrect in genus 1
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Reporter: dmharvey | Owner:
Type: defect | Status: new
Priority: major | Milestone: sage-6.9
Component: elliptic curves | Keywords:
Merged in: | Authors:
Reviewers: | Report Upstream: N/A
Work issues: | Branch:
Commit: | Dependencies:
Stopgaps: |
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The {{{cardinality_exhaustive}}} method can return incorrect results for
genus 1 curves if they are given by {{{y^2 = f(x)}}} where f(x) is a
quartic polynomial whose leading coefficient is not a square:
{{{
sage: ZZX.<x> = PolynomialRing(ZZ)
sage: f = 15*x^4 + 7*x^3 + 3*x^2 + 7*x + 18
sage: HyperellipticCurve(f.change_ring(GF(19))).cardinality_exhaustive(1)
20
sage: sum([legendre_symbol(u, 19) + 1 for u in [f(x) for x in range(19)] +
[f[4]]]) # correct answer
19
}}}
Here is the offending code:
{{{
if g == 1:
# elliptic curves always have one smooth point at infinity
a += 1
}}}
The problem is that for the given model, the curve may not have a rational
point at infinity.
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Ticket URL: <http://trac.sagemath.org/ticket/19122>
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