Re: Visibility of III[7]
William Stein
Sorry I didn't respond to this, because I've been pretty busy with an
undergrad REU research project this summer to find all the elliptic
curves over Q(sqrt(5)) with norm conductor up to 1831 (the first with
rank 2). We have all but *one* of the curves. I noticed that for
the one remaining curve E, there is another curve F that we know such
that E[7] = F[7]. This reminded me of your email. Just in case you
have any suggestions (see also the other email I cc'd you on, where we
worked out a strategy from your paper), here are the details.
Let a = (1+sqrt(5))/2. The curve F has a-invariants
[0,a-1,a+1,-2*a,0], i.e., it is:
y^2 + (a+1)*y = x^3 + (a-1)*x^2 + (-2*a)*x
which has conductor (3*a - 21) of norm 369. Our goal is to find a
curve E with conductor (-6*a+42) with E[7] = F[7].
I believe, because of computations I've done with Hilbert modular
forms, that there is such a curve E.
Any thoughts? Is all your Magma code for this sort of thing only
over QQ, rather than Q(sqrt(5)).
I also know that F has rank 1 and E has rank 0, if that matters.
Finally, E has a rational 2-torsion point, but no other torsion.
-- William