[with patch, with positive review] Bug in twisting for p_primary_bound on Tate-Shafarevich groups

[wuthrich]

William Stein found a bug in p_primary_part, namely

sage: E = EllipticCurve([-19,34]); E.cremona_label()  # y^2 = x^3 - 19*x + 34
'944e1'
sage: S = E.sha(); S
Shafarevich-Tate group for the Elliptic Curve defined by y^2 = x^3 -
19*x + 34 over Rational Field
sage: E.ap(5)
-3
sage: factor(944)
2^4 * 59
sage: S.an_padic(5)
Traceback (most recent call last):
...
ValueError: can not twist a curve of conductor (=472) by the quadratic
twist (=-4).

The problem is at 2 and 3, we have to check if we are allowed to twist.

And John Cremona suggested

Is it possible to add a doctest illustrating the suggestion to "try an_padic instead"? That would be useful for the reference manual.

[wuthrich]

I believe that the patch chooses now the correct twist.

William : could you use it for the table, before it goes in ? so that I am sure that there are no further problems with it. I have tested it only on a few examples.

[John Cremona]

Patch applies fine to 4.1.1, and tests run ok. The code looks ok to me too. I'm not quite expert enough to be 100% confident, but enough to pass this!