Implement division_field() for elliptic curves

[jdemeyer]

sage/schemes/elliptic_curves/gal_reps.py has code to compute a splitting field, but that clearly does not belong to elliptic curves. This is used to compute the division field. We should use the new splitting_field() function from #2217 to implement this. We should make it work also over number fields.

Depends on #2217
Depends on #15626
Depends on #11271

CC: @JohnCremona

Component: elliptic curves

Author: Jeroen Demeyer

Branch/Commit: u/jdemeyer/ticket/11905 @ bb38871

Reviewer: John Cremona

Issue created by migration from https://trac.sagemath.org/ticket/11905

[jdemeyer]

Dependencies: #11891

[jdemeyer]

Changed dependencies from #11891 to #11904, #11995, #2217

[chriswuthrich]

comment:4

Thanks for spotting and changing this. One could even argue that the division field should be available to all elliptic curves, not just the ones over QQ. But this could be changed later.

[jdemeyer]

comment:5

This is still very much work in progress by the way.

[jdemeyer]

comment:7

Replying to @categorie:

Thanks for spotting and changing this. One could even argue that the division field should be available to all elliptic curves, not just the ones over QQ. But this could be changed later.

I can easily do number fields, very general fields would be harder.

[jdemeyer]

Attachment: 11905_splitting_field.patch.gz

[fchapoton]

comment:9

#2217 is ready for review

[jdemeyer]

Author: Jeroen Demeyer

[jdemeyer]

Branch: u/jdemeyer/ticket/11905

[sagetrac-git]

Commit: a76a510

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. Recent commits:

a76a510	Implement division_field() for elliptic curves
c50eb3e	No need to specify caller_name in verbose()
6df69b6	Add comments, small cosmetic changes
9e2dd44	Make q monic before computing cubic resolvent
9b5558e	Implement splitting fields for number fields

[jdemeyer]

Changed dependencies from #11904, #11995, #2217 to #2217

[jdemeyer]

Changed dependencies from #2217 to #2217, #15626

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. Recent commits:

d6a7f3c	Implement division_field() for elliptic curves
776795d	Do polynomial consistency check only for minimal dm
df52508	Further improvements to splitting_field()
c50eb3e	No need to specify caller_name in verbose()
6df69b6	Add comments, small cosmetic changes

[sagetrac-git]

Changed commit from a76a510 to d6a7f3c

[sagetrac-git]

Changed commit from 810ceec to 3661e44

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

3661e44	Implement division_field() for elliptic curves
ca46e2b	Do polynomial consistency check only for minimal dm
7830eab	Further improvements to splitting_field()

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

9b406d0

Implement division_field() for elliptic curves

[sagetrac-git]

Changed commit from 3661e44 to 9b406d0

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

4d450f1

Implement division_field() for elliptic curves

[sagetrac-git]

Changed commit from 9b406d0 to 4d450f1

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

8b96090

Implement division_field() for elliptic curves

[sagetrac-git]

Changed commit from 4d450f1 to 8b96090

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

05c57d2

Implement division_field() for elliptic curves

[sagetrac-git]

Changed commit from 8b96090 to 05c57d2

[jdemeyer]

comment:28

Note that only 05c57d2 needs to be reviewed.

[JohnCremona]

comment:30

Looking good! Can we have an example where the X-division field equals the division field, i.e. where the final step is not necessary? And also an example (from the ones you give) showing that the splitting field of the division polynomial has half the degree of the division field? I think that this would be instructive to see in the reference manual. I see that in the code for the final step you again call splitting field with a quadratic; very efficient!

I think it says a lot for the good way in which you implemented splitting fields that this now becomes so simple!

[sagetrac-git]

Changed commit from 05c57d2 to bb38871

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

bb38871

division_field(): use map_coefficients() and add more examples

[JohnCremona]

Reviewer: John Cremona

[JohnCremona]

comment:32

I was expecting to be able to pull the new commit on top of the old one, but that failed (merge conflict I think). I just deleted the branch I had and made a new one.

Thanks for the new example, which do exactly what I wanted. I read through the code again, and like it, and tested all of Sage with this commit as well as some of my own examples.

[jdemeyer]

comment:33

Thanks again for the quick review!