#9417: Tamagawa number calculated incorrectly
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Reporter: arminstraub | Owner: cremona
Type: defect | Status: new
Priority: major | Milestone:
Component: elliptic curves | Keywords: tamagawa_number local_data
Author: | Upstream: N/A
Reviewer: | Merged:
Work_issues: |
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Changes (by cremona):
* cc: cturner, beankao (removed)
* cc: was, justin (added)
Comment:
As the author of both Sage's and Magma's code for Tamagawa numbers, I have
been tracking this one down. It turns out to be due to a bug in how
elements of the rings of integers are mapped into residue fields:
{{{
sage: K.<a> = NumberField(x^2+18*x+1)
sage: P = K.ideal(2)
sage: F = K.residue_field(P)
sage: R = PolynomialRing(F, 'x')
sage: R([0, a, a, 1])
x^3 + abar*x^2 + abar*x
sage: F(a)
1
sage: a.minpoly()
x^2 + 18*x + 1
sage: F.gen()
abar
sage: F.gen().minpoly()
x^2 + x + 1
}}}
The polynomial {{{x^3+a*x^2+a*x}}} reduced modulo P=(2) wrongly to
{{{x^3+abar*x^2+abar*x}}}. Although the generator of the residue field F
is suggestively called abar, it it *not* the reduction of a mod P (which
is 1 mod P).
I will open a new ticket for that, and try to fix it. This ticket can
probably then be closed, so watch this space.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/9417#comment:1>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
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