1476 found !!!!!!!!!!!!!!!!!!!!!!!!!!!!!
William Stein
The missing curve of norm conductor 1476 is:
E = EllipticCurve([1,-a-1,a+1, 3904*a-6577, 53806*a-88477])
!!!!!!
It's canonical minimal model is
[a, a - 1, 0, -257364*a - 159063, -75257037*a - 46511406]
We thus now have *all* curves of norm conductor <= 1831 = norm
conductor of first curve of rank 2. Just in time, since today is the
last day of the REU.
See the email below from Tom Fisher. At the last moment yesterday I
wrote to him explaining my idea for finding the 1476 curve and that we
were going to try it.... but wondered if he could. Much to my very
pleasant surprise, Tom was quickly able to adapt his code (over Q) to
Q(sqrt(5)) and use it to find the missing curve. Again, a quick
summary of the technique is that (1) our mystery curve E has the same
a_p modulo 7 (i.e., the same E[7]) as a curve of norm conductor 369
that we know, and (2) one can search for all curves with a given E[7]
once you have one of them by searching for points on a quartic
surface.
-- William
---------- Forwarded message ----------
From: T.A. Fisher <T.A.F...@dpmms.cam.ac.uk>
Date: Fri, Aug 12, 2011 at 5:58 AM
Subject: Re: Visibility of III[7]
To: William Stein <wst...@gmail.com>
Cc: uwntr...@googlegroups.com
Dear William,
I've been working more on the visibility of III[7], and now have
six examples of pairs (E,C) where E is an elliptic curve over Q,
C is a genus 2 curve over Q, and I can prove III(E) has a subgroup
(Z/7Z)^2 visible in an abelian 3-fold isogenous to E x Jac(C).
I'll send you some notes on this in the next couple of weeks.
I'd be interested if you think the modular symbols calculations
could be pushed further.
The formula for X_E(7) is due to Halberstadt and Kraus, and the
one for X_E^-(7) was deduced from it by Poonen--Schaefer--Stoll.
I gave new proofs in my preprint for n=9,11. I also wrote a
program to minimise and reduce these curves, but only over Q.
This morning I just hacked the minimisation to work over Q(sqrt{5}),
and did something very ad hoc for the reduction. The points
on X_E(7) were rather small, so it might have been possible to
succeed without first minimising and reducing (but I didn't
know that before I tried).
Here are your missing curves (phi = a):
E1 := EllipticCurve([ 1, -phi - 1, phi + 1,
3904*phi - 6577, 53806*phi - 88477 ]);
E2 := EllipticCurve([ phi, phi, phi + 1,
-5179442*phi - 3201071, -6788348449*phi - 4195430069 ]);
I've attached a Magma file with a few more details. (N.B. there
is also a Magma file with all the formulae on my webpage, so you
shouldn't have needed to type them in!)
Best wishes,
Tom
On Thu, 11 Aug 2011, William Stein wrote:
> Hi Tom,
>
> 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
>
--
William Stein
Professor of Mathematics
University of Washington
http://wstein.org
Ben LeVeque
> <forwilliam.m>
Jonathan Bober
Awesome!
I tried finding the other missing curve a bit (using ssh on my phone) but have had no luck. It's difficult using my phone, though, so I haven't looked too hard.
William Stein
> Awesome!
>
> I tried finding the other missing curve a bit (using ssh on my phone) but
> have had no luck. It's difficult using my phone, though, so I haven't looked
> too hard.
This is a good reason to stop our official table at 1831, completing
"phase 2". :-).
We can have a "phase 3" project, which will be to get to the first
curve of rank 3 (around 25000, probably). This will be in the NSF
grant proposal I'm writing in the next few weeks...
-- William