#9320: Implement root numbers for elliptic curves over number fields
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Reporter: arminstraub | Owner: cremona
Type: enhancement | Status: needs_work
Priority: minor | Milestone:
Component: elliptic curves | Resolution:
Keywords: root number | Merged in:
Authors: Tim Dokchitser | Reviewers:
and group (Sage Days 22) | Work issues: fix ReST formatting,
Report Upstream: N/A | coverage
Branch: u/chapoton/9320 | Commit:
Dependencies: | 87938e0bdccc397f9527b1db39b9c85006f40232
| Stopgaps:
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Comment (by cremona):
I suggest that a good source of examples would be elliptic curves over
number fields where we know the associated modular form, since the root
number at a bad prime should match the Atkin-Lehner eigenvalue. (The
alternative would be to compue a whole lot of examples with Magma, but
that would make me uncomfortable; nevertheless we should of course check
that our results are compatible with Magma.)
There is no issue when the primes have multiplicative reduction, since
then the root number is very easy being minus E.ap, i.e. depends only on
whether the number of points on the reduction is p+1 or p-1 (of course "p"
means Norm(p) in the number field case). It's the case of additive
reduction at primes dividing 2 or 3 which are harder.
Here is one taken from my thesis (see http://www.numdam.org/numdam-
bin/search?h=nc&id=CM_1984__51_3_275_0&format=complete):
{{{
K.<i>=QuadraticField(-1)
E=EllipticCurve([1+i, 1+i, 0,i,0])
P2=K.ideal(i+i)
E.root_number(P2)
-1
}}}
which I checked with Magma. The conductor here is {{{P2^2 * P41}}}.
Is this what you want? How many examples do you need?
Tables of elliptic curves over number fields do exist, and were in fact
one of the topics of last week's Curves and Automorphic Forms workshop in
Arizona.
--
Ticket URL: <http://trac.sagemath.org/ticket/9320#comment:13>
Sage <http://www.sagemath.org>
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