Hello,
I recently discovered that the following command does not work:
sage: EllipticCurve('390g1').quadratic_twist(-191).gens()
The reason is that Sage tries to find generators using mwrank, and the
height bounds Sage uses by default are set too low:
-------------------------------------------------------
Summary of results:
-------------------------------------------------------
0 <= rank(E) <= 1
#E(Q)/2E(Q) >= 2
Information on III(E/Q):
#III(E/Q)[phi'] <= 2
#III(E/Q)[2] <= 2
Information on III(E'/Q):
#phi'(III(E/Q)[2]) = 1
#III(E'/Q)[phi] = 1
#III(E'/Q)[2] <= 2
The following false advice appears in the docstring:
{{{
HINT: If you would like to control the height bounds used in the
2-descent, first call the two_descent function with those height
bounds. However that function, while it displays a lot of output,
returns no values.
}}}
The problem with this is that E.two_descent does not cache any
information it computes, so this really doesn't help.
Now that mwrank is available as a library, I'd like to clean up a few
places in ell_rational_field.py where Sage is using mwrank. I thought
it would be a good idea to open a thread on sage-devel first. I'd like
to do the following:
1. E.two_descent should call the mwrank library instead of the shell,
and should cache everything it computes.
2. E.simon_two_descent should cache exactly the same data.
3. E.rank should use the library instead of the shell.
4. E.gens should use the library instead of the shell.
5. E.gens should simply run indefinitely if sha[2]_an is trivial, or
at least have an option to this effect. Perhaps E.regulator should use
this option, so that E.heegner_index never blows up (right now, it can
fail if the height bounds are not high enough for the curve).
Please comment on these ideas, I'm hoping to use this thread as a
place to organize my thoughts before I start doing the work.
--
Robert L. Miller
http://www.rlmiller.org/
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