#12509: computation of height of point on elliptic curve over Q(sqrt(5)) is
WRONG
-----------------------------------+----------------------------------------
Reporter: was | Owner: was
Type: defect | Status: needs_review
Priority: critical | Milestone: sage-5.6
Component: elliptic curves | Resolution:
Keywords: heights | Work issues:
Report Upstream: N/A | Reviewers:
Authors: John Cremona | Merged in:
Dependencies: | Stopgaps: #12692
-----------------------------------+----------------------------------------
Changes (by {'newvalue': u'John Cremona', 'oldvalue': ''}):
* keywords: => heights
* status: new => needs_review
* component: number theory => elliptic curves
* author: => John Cremona
Old description:
> There are evidently many examples in which computing {{{P.height()}}},
> for {{{P}}} a point on an elliptic curve over Q(sqrt(5)) yields a
> completely wrong answer. This is very serious, since it is a blatantly
> wrong mathematical answer -- raising NotImplementedError would be much
> better! Here's an example that Ashwath Rabindranath (Princeton) found,
> where Sage and Magma do not agree. According to BSD, Sha has order 1
> using the Magma answer, and a crazy order with the Sage answer.
>
> {{{
> sage: K.<a> = NumberField(x^2-x-1)
> sage: v = [0, a + 1, 1, 28665*a - 46382, 2797026*a - 4525688]
> sage: E = EllipticCurve(v)
> sage: E == E.global_minimal_model()
> True
> sage: F.a_invariants()
> (0, a + 1, 1, 28665*a - 46382, 2797026*a - 4525688)
> sage: P = E([72*a - 509/5, -682/25*a - 434/25])
> sage: P.height()
> 1.35648516097058
> sage: Q = magma(E)(magma([P[0], P[1]]))
> sage: Q
> (1/5*(360*a - 509) : 1/25*(-682*a - 434) : 1)
> sage: Q.Height()
> 1.38877711688727252538242306
> }}}
New description:
There are evidently many examples in which computing {{{P.height()}}}, for
{{{P}}} a point on an elliptic curve over Q(sqrt(5)) yields a completely
wrong answer. This is very serious, since it is a blatantly wrong
mathematical answer -- raising NotImplementedError would be much better!
Here's an example that Ashwath Rabindranath (Princeton) found, where Sage
and Magma do not agree. According to BSD, Sha has order 1 using the Magma
answer, and a crazy order with the Sage answer.
{{{
sage: K.<a> = NumberField(x^2-x-1)
sage: v = [0, a + 1, 1, 28665*a - 46382, 2797026*a - 4525688]
sage: E = EllipticCurve(v)
sage: E == E.global_minimal_model()
True
sage: F.a_invariants()
(0, a + 1, 1, 28665*a - 46382, 2797026*a - 4525688)
sage: P = E([72*a - 509/5, -682/25*a - 434/25])
sage: P.height()
1.35648516097058
sage: Q = magma(E)(magma([P[0], P[1]]))
sage: Q
(1/5*(360*a - 509) : 1/25*(-682*a - 434) : 1)
sage: Q.Height()
1.38877711688727252538242306
}}}
Apply: trac12509-heights.patch
--
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/12509#comment:9>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
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