#7935: local_data for elliptic curves over number fields
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Reporter: wuthrich | Owner: cremona
Type: defect | Status: needs_review
Priority: major | Milestone: sage-4.3.1
Component: elliptic curves | Keywords: elliptic curve, number fields,
local data, tamagawa
Work_issues: | Author:
Upstream: N/A | Reviewer:
Merged: |
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Comment(by wuthrich):
Maybe you make a little sign error ? In the bug-example, the valuation of
a6 at P = (a) is -2. So e is -1. So you are really multiplying the ais by
pi^+i^. So pi should be integral away from P.
As to your comment on {{{tamagawa_product()}}}, we have
{{{
sage: E = EllipticCurve('11a2')
sage: E.tamagawa_product()
1
sage: E2 = E.change_weierstrass_model([1/11,0,0,1])
sage: E2
Elliptic Curve defined by y^2 + 3993*y = x^3 - 121*x^2 - 114492620*x -
466951591502 over Rational Field
sage: E2.tamagawa_product()
1
}}}
So it is defined to be the product of the Tamagawa numbers (i.e. the index
of E^0^ (K_v)) on the minimal model.
My definition (and that is the only sensible over number fields) of
{{{tamagawa_product()}}} changes as one changes the chosen invariant
differential in such a way that the product of it with the periods is
invariant under this choice. In particular it is not the product of the
Tamagawa numbers despite the name.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/7935#comment:6>
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