#8451: improve galois representation for elliptic curves
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Reporter: wuthrich | Owner: cremona
Type: defect | Status: needs_review
Priority: major | Milestone: sage-4.4
Component: elliptic curves | Keywords: elliptic curves, galois
representation, is_surjective
Author: Chris Wuthrich | Upstream: N/A
Reviewer: | Merged:
Work_issues: |
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Comment(by was):
From Drew Sutherland, who is an expert on this algorithm:
"Hi William,
I am in Montreal at the moment preparing for a talk, but I would be happy
to take a look at it this weekend when I should have some time. One thing
that would be helpful is to be able to see the new version of
sage/schemes/elliptic_curves/gal_reps.py
in its entirety. Currently, I can browse the old version and look at the
diff, but it would be easier to just read linearly through the new
version.
A very useful test would be to run the new code on some substantial subset
of the Cremona database and compare the results to the answer you get from
my code. Is this something that can easily be done now that you have
integrated my code into Sage?
Drew
p.s. I can say that I agree with the comment that the mod 3 rep is
surjective if and only if the 3-division poly has Galois group S_4.
Another way to see this is to look at the Galois group of the instantiated
modular polynomial Phi_3(X,j(E)), which happens to be isomorphic to the
Galois group of the 3-division poly (3 is special in this regard) and is
also isomorphic to the image of the mod 3 Galois rep in PGL(2,3)."
My response, is that yes, I'm sure we can set you up with everything you
want above.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/8451#comment:7>
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