implement regulator function for elliptic curves over number fields

[JohnCremona]

Now that we have canonical heights over number fields, the regulator_of_points code can be moved up from ell_rational_field to ell_number_field.

CC: @robertwb

Component: elliptic curves

Author: John Cremona

Reviewer: David Loeffler, Robert Bradshaw

Merged: sage-4.5.2.alpha0

Issue created by migration from https://trac.sagemath.org/ticket/9372

[JohnCremona]

Applies to 4.4.4

[JohnCremona]

comment:2

Attachment: trac_9372-regulator.patch.gz

The patch moves the two functions height_pairing_matrix and regulator_of_points, and adds doctests over number fields.

[JohnCremona]

Author: John Cremona

[loefflerd]

comment:3

I'm getting a couple of doctest failures -- something is off by a sign in the height_pairing_matrix code:

sage -t -long ell_number_field.py
**********************************************************************
File "/storage/masiao/sage-4.4.4/devel/sage-reviewing/sage/schemes/elliptic_curves/ell_number_field.py", line 308:
    sage: E.height_pairing_matrix([P,Q])
Expected:
    [ 0.686667083305587 -0.268478098806726]
    [-0.268478098806726  0.327000773651605]
Got:
    [0.686667083305587 0.268478098806726]
    [0.268478098806726 0.327000773651605]
**********************************************************************
File "/storage/masiao/sage-4.4.4/devel/sage-reviewing/sage/schemes/elliptic_curves/ell_number_field.py", line 317:
    sage: EK.height_pairing_matrix([EK(P),EK(Q)])
Expected:
    [ 0.686667083305586 -0.268478098806726]
    [-0.268478098806726  0.327000773651605]
Got:
    [0.686667083305586 0.268478098806726]
    [0.268478098806726 0.327000773651605]
**********************************************************************
1 items had failures:
   2 of  23 in __main__.example_4
***Test Failed*** 2 failures.

Also, a very tiny quibble: the second argument "precision" to height_pairing_matrix is missing its bullet point in the docstring.

[JohnCremona]

comment:4

Probably I used E.gens() which is a bad idea in odctests, better to enter the points manually.

No time to fix now, about to leave SD22 for home.... but thanks all the same! Put this patch up from a coffee shop last night just before it closed (about a dozen Sagers being chased out!)

[adeines]

comment:5

@JohnCremona: You did you E.gens(). I get:

sage: E = EllipticCurve('389a1')
sage: E.gens()
[(-1 : 1 : 1), (0 : -1 : 1)]

Should I change the doc test to the following?

sage: E = EllipticCurve('389a1')
sage: P,Q = E.point([-1,1,1]),E.point([0,-1,1])
sage: E.height_pairing_matrix([P,Q])
[0.686667083305587 0.268478098806726]
[0.268478098806726 0.327000773651605]

}}}

[adeines]

doctest fixed -- replaces previous patch

[adeines]

comment:6

Attachment: trac_9372-regulator.2.patch.gz

If the only problem was that the doctest called E.gens(), this fixes those doctests and you can give this a positive review.

[JohnCremona]

comment:8

Thanks to all for sorting that out. I should have known better. Even for curves in the database, it is not safe to use gens() since unless you have the larger database installed the gens are computed on the fly and are not unique. (And doctests definitely should not assume an optional spkg is installed!).

[qed777]

Reviewer: David Loeffler, Robert Bradshaw

[qed777]

Merged: sage-4.5.2.alpha0

[qed777]

comment:9

I'm updating the Reviewer(s) field. Please correct me if I'm wrong.