#12509: computation of height of point on elliptic curve over Q(sqrt(5)) is
WRONG
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Reporter: was | Owner: was
Type: defect | Status: new
Priority: critical | Milestone: sage-5.6
Component: number theory | Resolution:
Keywords: | Work issues:
Report Upstream: N/A | Reviewers:
Authors: | Merged in:
Dependencies: | Stopgaps: #12692
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Comment (by cremona):
It seems to be a numerical stability problem. With the same example, and
using the first embedding of K, we have, first to default 53-bit
precision:
{{{
sage: b2,b4,b6,b8
(1.527864045, -128195.888575, -25017379.5417, -4118102250.56)
sage: x
-146.29844719
sage: t
-0.00683534254264
sage: 1 - (b4 * t**2) - (2*b6 * t**3) - (b8 * t**4)
3.5527136788e-15
}}}
while at 100-bit precision:
{{{
sage: bb2,bb4,bb6,bb8
(1.5278640450004206071816526625, -128195.88857503147164756896321,
-2.5017379541668653550777130780e7, -4.1181022505609323099570853156e9)
sage: xx
-146.29844718999242907073025207
sage: tt
-0.0068353425426405016233236537484
sage: 1 - (bb4 * tt**2) - (2*bb6 * tt**3) - (bb8 * tt**4)
3.5142315895588997690642550530e-15
}}}
This means that whenever we evaluate one of the four polynomials (which
coefficients are essentially the b-invariants) at some point we may lose
precision badly.
One way out would be to use a "working precision" higher than the
requested precision. Another would be to use algebraic values instead of
approximate ones for t,z,w in the algorithm, which is likely to be slow,
but I will try it.
A better solution for the longer term would be to implement Mestre's AGM
algorithm instead BUT this only exists at present for real embeddings (I
have worked on extending to the complex case, but that is still in
progress).
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/12509#comment:7>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
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