aplists up to 50,000 for two curves

[William Stein]

Hi Jon,

You can now try using your algorithm to much higher precision, since I
was able to compute the aplists to 50000 on two curves!!

N = -38*a + 26
http://wstein.org/home/wstein/reu/2011/aplist_1756_50020.sobj

N = -40*a + 24
http://wstein.org/home/wstein/reu/2011/aplist_1984_49800.sobj

William

--
William Stein
Professor of Mathematics
University of Washington
http://wstein.org

[William Stein]

Hi,

I'm also computing the aplists for all the remaining curves here:

http://wiki.sagemath.org/reu/2011/schedule?action=AttachFile&do=view&target=Curves_MISSING_8-9-11.txt

I don't know how many of these are already known, since Ben did find
several new ones yesterday, but didn't update the list...

william

[Ben LeVeque]

Hi,

   Right, I will post a new list in a few minutes! Any luck finding the curves?

-Ben

[William Stein]

On Wed, Aug 10, 2011 at 10:52 AM, Jonathan Bober <jwb...@gmail.com> wrote:
>
>
> On Wed, Aug 10, 2011 at 9:33 AM, William Stein <wst...@gmail.com> wrote:
>>

>> Hi Jon,
>>
>> You can now try using your algorithm to much higher precision, since I
>> was able to compute the aplists to 50000 on two curves!!
>>
>> N = -38*a + 26
>>   http://wstein.org/home/wstein/reu/2011/aplist_1756_50020.sobj
>>
>

> This should be it:
>
> Trying c4 = 48019*a + 30154 ; c6 =  -15835084*a - 9796985  ; D = 192*a + 608
> Elliptic Curve defined by y^2 + a*x*y + a*y = x^3 + (a-1)*x^2 +
> (-1001*a-628)*x + (17899*a+11079) over Number Field in a with defining
> polynomial x^2 - x - 1

WOW!!!! Look how big those coefficients are. Here's the
Ohana-canonical minimal model a-invariants -- hard to find with a
straightforward search:

(a, a - 1, a, -1001*a - 628, 17899*a + 11079)

Did you use and need my enhanced higher-precision aplist?

sage: E.torsion_order()
1

[William Stein]

On Wed, Aug 10, 2011 at 11:19 AM, Jonathan Bober <jwb...@gmail.com> wrote:

>
>
> On Wed, Aug 10, 2011 at 11:13 AM, William Stein <wst...@gmail.com> wrote:
>>
>> On Wed, Aug 10, 2011 at 10:52 AM, Jonathan Bober <jwb...@gmail.com>
>> wrote:
>> >
>> >
>> > On Wed, Aug 10, 2011 at 9:33 AM, William Stein <wst...@gmail.com> wrote:
>> >>
>> >> Hi Jon,
>> >>
>> >> You can now try using your algorithm to much higher precision, since I
>> >> was able to compute the aplists to 50000 on two curves!!
>> >>
>> >> N = -38*a + 26
>> >>   http://wstein.org/home/wstein/reu/2011/aplist_1756_50020.sobj
>> >>
>> >
>> > This should be it:
>> >
>> > Trying c4 = 48019*a + 30154 ; c6 =  -15835084*a - 9796985  ; D = 192*a +
>> > 608
>> > Elliptic Curve defined by y^2 + a*x*y + a*y = x^3 + (a-1)*x^2 +
>> > (-1001*a-628)*x + (17899*a+11079) over Number Field in a with defining
>> > polynomial x^2 - x - 1
>>
>> WOW!!!!  Look how big those coefficients are.   Here's the
>> Ohana-canonical minimal model a-invariants -- hard to find with a
>> straightforward search:
>>
>>      (a, a - 1, a, -1001*a - 628, 17899*a + 11079)
>>
>> Did you use and need my enhanced higher-precision aplist?
>>
>

> I don't know exactly what I needed, but with the higher precision list I
> found the curve in a few seconds.

Very, very nice. That's incredible. It would be useful to know what
precision is needed.

For all the remaining curves, I've computed the aplists for primes up
to norm conductor about 12000 now, which took about 45 minutes,
including loading pickles into memory.

Somebody should do some estimates about the feasibility of
systematically pushing the table of elliptic curves up to norm
conductor 26,569, which is where the first (known) rank 3 curve first
appears...

-- William

> For it to be only a few seconds, it still
> required me to look at a few twists by hand to determine which multiples of
> the periods the first twists were. (36, 5, 5, and 1 to start, at least.) I
> had done that last night, but there still didn't seem to be enough
> precision.

[Ben LeVeque]

Nice!! That's great!! Here's an updated list of missing curves, as of a few minutes ago at least:

http://wiki.sagemath.org/reu/2011/schedule?action=AttachFile&do=view&target=Curves_Missing_8-11-10.txt

I'll update the list in another hour or two, to include the Bobele curves.

[Ben LeVeque]

Also, just for comparison, ModJon had been running on that curve for probably over a day!

[William Stein]

On Wed, Aug 10, 2011 at 11:29 AM, Ben LeVeque <ben.l...@gmail.com> wrote:

> Nice!! That's great!! Here's an updated list of missing curves, as of a few
> minutes ago at least:
> http://wiki.sagemath.org/reu/2011/schedule?action=AttachFile&do=view&target=Curves_Missing_8-11-10.txt
> I'll update the list in another hour or two, to include the Bobele curves.

So we now have exactly six curves left in order to reach 2000, and
exactly 3 curves left in order to be complete up to the first curve of
rank 2 (!).

1475 35*a - 25 [-1,'?',-1,-1,-1,-8,-7,9,-9,1,-8,6,11,-5,'?','?',-2,-11,9,6,10,-3,-10,0]
1475 -35*a + 25
[1,'?',1,-1,-1,8,7,-9,9,1,-8,6,11,5,'?','?',-2,-11,9,6,-10,3,10,0]
1476 -6*a + 42 ['?',4,'?',0,2,-4,6,-8,-2,-8,-8,'?','?',-10,-14,6,8,-8,12,8,8,16,6,-14]
1856 -8*a + 48 ['?',-1,4,6,1,-5,8,'?','?',-7,-4,1,0,1,12,-4,8,6,5,8,10,1,-17,-8]
1856 -8*a + 48 ['?',-4,4,-2,4,-4,0,'?','?',4,4,10,8,-2,-10,12,-6,4,10,0,16,-14,6,14]
1856 -8*a + 48 ['?',3,4,-2,-3,3,0,'?','?',-3,4,-11,8,5,4,12,8,-10,-11,0,2,-7,-1,0]

[Ben LeVeque]

1984 was found as well? Great!!

[Jonathan Bober]

There's something funny there in the first two curves. They have the same conductor and appear to be twists of each other.

The conductor factors as (2*a - 1)^2 * (7*a - 5) = 5 * (7*a - 5). Where is the list of curves that we know about?

[Ben LeVeque]

Hmm, that's true. Very similar aplists... 

http://wiki.sagemath.org/reu/2011/schedule?action=AttachFile&do=view&target=Curves_2000_Found_8-10-11.txt

[Ben LeVeque]

Those are just the curves found since last week, though -- there are a lot more that were found by Joanna. I will try to find the list of those.

[Ben LeVeque]

This is the original matched.txt file of curves found by Joanna

[Jonathan Bober]

I wanted to look at a list like that for all the curves. In particular, I wanted to look at things with conductors which divide 35*a - 25. But that probably won't help (and you already did that). Perhaps they are twists of one another by (2*a - 1).

[Jonathan Bober]

I think they are twists of each other by (2 * a - 1).

sage: from psage.modform.hilbert.sqrt5.ellcurve import *       
sage: primes = primes_of_bounded_norm(100)
sage: [chi(p.sage_ideal()) for p in primes]
[-1, 0, -1, 1, 1, -1, -1, -1, -1, 1, 1, 1, 1, -1, -1, -1, 1, 1, 1, 1, -1, -1, -1, -1]

Compare that to:

1475 35*a - 25 [-1,'?',-1,-1,-1,-8,-7,9,-9,1,-8,6,11,-5,'?','?',-2,-11,9,6,10,-3,-10,0]
1475 -35*a + 25 [1,'?',1,-1,-1,8,7,-9,9,1,-8,6,11,5,'?','?',-2,-11,9,6,-10,3,10,0]

and you can see that the sign change is exactly the sign of the character.

So we only need to find one of those curves.

[Ben LeVeque]

Oh, cool! Are the aplists at a point where Dembele could start running on one of them?

[William Stein]

On Wed, Aug 10, 2011 at 12:19 PM, Ben LeVeque <ben.l...@gmail.com> wrote:
> Oh, cool! Are the aplists at a point where Dembele could start running on
> one of them?
>

It's up to norm conductor 17000. It'll be at least to 20000 when I
arrive at the office (leaving in a few minutes).

I have to compute the a_p for p=59a (or 59b) -- the bad prime -- "by
hand" still.

For 1475 with 35*a-25 we have:

sign = -1
ord_{s=1} L(E,s) = 1
a_{59a} = -1

and here it is (up to 18000 or so):

http://wstein.org/home/wstein/reu/2011/aplist-1475-2.sobj

Hopefully this is enough precision, though I wouldn't be surprised if it isn't.

-- William

[William Stein]

On Wed, Aug 10, 2011 at 11:38 AM, William Stein <wst...@gmail.com> wrote:
> On Wed, Aug 10, 2011 at 11:29 AM, Ben LeVeque <ben.l...@gmail.com> wrote:
>> Nice!! That's great!! Here's an updated list of missing curves, as of a few
>> minutes ago at least:
>> http://wiki.sagemath.org/reu/2011/schedule?action=AttachFile&do=view&target=Curves_Missing_8-11-10.txt
>> I'll update the list in another hour or two, to include the Bobele curves.
>
> So we now have exactly six curves left in order to reach 2000, and
> exactly 3 curves left in order to be complete up to the first curve of
> rank 2 (!).
>
> 1475 35*a - 25 [-1,'?',-1,-1,-1,-8,-7,9,-9,1,-8,6,11,-5,'?','?',-2,-11,9,6,10,-3,-10,0]
> 1475 -35*a + 25
> [1,'?',1,-1,-1,8,7,-9,9,1,-8,6,11,5,'?','?',-2,-11,9,6,-10,3,10,0]

All four of the following have sign=1:

> 1476 -6*a + 42 ['?',4,'?',0,2,-4,6,-8,-2,-8,-8,'?','?',-10,-14,6,8,-8,12,8,8,16,6,-14]

http://wstein.org/home/wstein/reu/2011/aplist-1476.sobj

> 1856 -8*a + 48 ['?',-1,4,6,1,-5,8,'?','?',-7,-4,1,0,1,12,-4,8,6,5,8,10,1,-17,-8]

http://wstein.org/home/wstein/reu/2011/aplist-1856a.sobj

> 1856 -8*a + 48 ['?',-4,4,-2,4,-4,0,'?','?',4,4,10,8,-2,-10,12,-6,4,10,0,16,-14,6,14]

http://wstein.org/home/wstein/reu/2011/aplist-1856b.sobj

> 1856 -8*a + 48 ['?',3,4,-2,-3,3,0,'?','?',-3,4,-11,8,5,4,12,8,-10,-11,0,2,-7,-1,0]

http://wstein.org/home/wstein/reu/2011/aplist-1856c.sobj

[Jonathan Bober]

These are the curves I wrote on the board earlier.

Elliptic Curve defined by y^2 + x*y = x^3 + (a-1)*x^2 + (1094*a-1770)*x + (-21331*a+34516)

Elliptic Curve defined by y^2 = x^3 + (108297*a-175311)*x + (20509200*a-33215130) 

[Jonathan Bober]

> 1856 -8*a + 48 ['?',-4,4,-2,4,-4,0,'?','?',4,4,10,8,-2,-10,12,-6,4,10,0,16,-14,6,14]

Elliptic Curve defined by y^2 = x^3 + (a+1)*x^2 + (-1439*a-893)*x + (-32764*a-20247) over Number Field in a with defining polynomial x^2 - x - 1

[Ben LeVeque]

Awesome!!!

[Jonathan Bober]

> 1856 -8*a + 48 ['?',3,4,-2,-3,3,0,'?','?',-3,4,-11,8,5,4,12,8,-10,-11,0,2,-7,-1,0]

 Elliptic Curve defined by y^2 = x^3 + (a+1)*x^2 + (-78*a-65)*x + (401*a+219) over Number Field in a with defining polynomial x^2 - x - 1

 With this one the discriminant is -26624*a - 18432 = (-3*a - 2) * 2^11 * (-a + 6). With such a high power of 2 there, we need to do a lot of guessing of discriminants before we reach it, so it is important to recognize early on that we need to divide the first mixed period guess by 9 and the second by 49 if we want to reach the right curve in a reasonable amount of time.

[William Stein]

Hi,

To get to the first rank 2 we just need to deal with this one:

1476 -6*a + 42
['?',4,'?',0,2,-4,6,-8,-2,-8,-8,'?','?',-10,-14,6,8,-8,12,8,8,16,6,-14]

I updated the aplist file for the above curve:

http://wstein.org/home/wstein/reu/2011/aplist-1476.sobj

It now includes all a_P with Norm(P) < 42491.

-- William

[Jonathan Bober]

I still can't get it to work on this curve. Something funny seems to be happening. Are we sure that it's a newform?

I pushed my latest changes to my googlecode psage clone. There is a lot of mess there, but the most relevant function is:

def find_curve_from_Lfunction(level, aplist, aplist_size, sign, ncpus = 1, limit1=1, limit2 = (0,5), limit3 = (0,5), limit4 = 0, verbose=0, chi_norm_bound = 1000, implementation = "cython"):

    """

    Given some $L$-function data, find an elliptic curve over $\Q(\sqrt 5)$

    with this $L$-function. Uses the method described in Lassina Dembele's

    paper "..."

    This method has not really been tested enough yet to know what the best

    way to use it is. In general, if we know a lot ap values, then it should

    be a good thing to set limit1 to be something like 4 or 5, or maybe even

    higher. However, if we don't know that many coefficients, setting limit1

    this high will causes problems because of precision issues. The reason

    for setting limit1 higher, though, is so that limit3 can be set lower, so

    that a much smaller space needs to be searched.

    In principal, one should be able to just so something like

    E = find_curve_from_Lfunction(level, aplist, aplist_size, sign)

    (with those variables set appropriately, of course.) If it doesn't work,

    you can try again with limit2 and/or limit3 set to a larger range. If

    the ranges are large enough and mixed periods were computed to enough

    precision, then the curve should be found. If there isn't enough

    precision, then it might be necessary to try increasing limit4. (Or

    you might roll some dice and use those to pick coefficients

    of an elliptic curve. Maybe it will be the one you are looking for!)

    In practice, it is probably best to take limit1 to be something between

    3 and 10, set verbose = 1, and look at the output to make sure that

    nothing bad happened. If the code is having trouble automatically

    identifying rational numbers because of limited precision, you can look

    at the output and try to do it manually, and then use the function

    find_curve_from_period_guesses(). Or you can just set limit1 to be lower

    and set limit3 higher, and let the computer do the work.

    INPUT::

        level - An ideal. The conductor of the curve/the level of the Hilbert

            modular form. Must not be a square.

        

        aplist - A list of Fourier coefficients of the modular form at primes.

            The nth entry in this list should correspond to the nth entry in

            primes_of_bounded_norm.

        aplist_size - The norm of prime corresponding the the last entry in aplist

        sign - The sign of the functional equation

        chi_norm_bound - the largest norm conductor we are willing to twist by

        ncpu - The number of cpus to use while computing twists of the L-function

        limit1 - the maximum number of twists we compute for each mixed period

        limit2 - a range on the powers of the discriminant we are going to

            try. We will try all possibilities D = level * prod(p_k^e_k) * u

            where p_k are the divisors of the level and sum e_k is in the range

            limit2

        limit3 - a range of the integers we will try dividing the mixed period

            guesses by. After computing initial guesses, we will try guesses

            of the form (\Omega_k)/m_k where \sum m_k is in the range limit3

        limit4 - only used in the sage implementation of searching after

            computing periods. After computing an initial guess of

            c4 = A + B \varphi, we will try all possibilities in the range

            (A += limit4), (B += limit4). This greatly slows things down

            (by a factor of 4 limit4^2, and probably shouldn't be necessary

            if we have enough precision

        implementation - 'sage' or 'cython'. The code we will use for

            searching the space of possible guesses. The cython implemenation

            is much faster, but has some overflow issues which may make it

            miss things that it should find. (But I don't know yet of an

            example where this happens. It is possible that when we have

            such large numbers we shouldn't be trying to find the curve

            anyway.)

    EXAMPLE::

        

        A simple example from a curve that we know about comes first.

        sage: from psage.modform.hilbert.sqrt5.ellcurve import *

        sage: E = EllipticCurve(K, [0, -a, a, 0, 0])

        sage: E2 = find_curve_from_Lfunction(E.conductor(), compute_aplist(E, 40000), 40000, 1, ncpus=4, limit1=3, verbose=0) # not really random output, but too much right now

        

        The curve we get back is in fact isomorphic to the original, but in general that is asking too much,

        so we just check isogeny.

        sage: E.is_isogenous(E2)

        True

        Here is a nice example that works well, which relies on a bunch of Fourier coefficients that

        William Stein computed for a Hilbert modular form of level 48 - 8*a.

        sage: aplist = load("http://wstein.org/home/wstein/reu/2011/aplist-1856b.sobj") # some random message about loading gets printed here

        sage: level = K.ideal(48 - 8*a)

        sage: sign = 1

        sage: E = find_curve_from_Lfunction(level, aplist, 29000, sign, ncpus=4, limit1=3, limit2=(0, 10), limit3=(0, 5), limit4 = 0, verbose=1) # again, not really random, but noisy

        sage: E

        Elliptic Curve defined by y^2 = x^3 + (a+1)*x^2 + (-1439*a-893)*x + (-32764*a-20247) over Number Field in a with defining polynomial x^2 - x - 1

        sage: E.conductor() == level

        True

        sage: x = compute_aplist(E, 100); x == aplist[:len(x)]

        True

    """

It should hopefully be easy to use, and those doctests actually pass, and don't take that long to run. The function compute_period_guesses() can also be useful for digging deeper when it doesn't work as well as it should.

I think the code will run with no changes to sage. If I made any changes that it relies on, then they should be in my sage directory on sage.math.washington.edu, so you can get them by pointing mercurial to

ssh://sage.math.washington.edu//home/bober/sage/devel/sage

The only patch applied on my home machine that isn't applied there is William's recent fix for number fields not being unique. That should be irrelevant.

[R. Andrew Ohana]

Just checked, it is indeed a newform.

--

Andrew

[William Stein]

Paul, see below for 1984.

---------- Forwarded message ----------
From: Jonathan Bober <jwb...@gmail.com>
Date: Wed, Aug 10, 2011 at 11:15 AM
Subject: Re: aplists up to 50,000 for two curves
To: William Stein <wst...@gmail.com>

On Wed, Aug 10, 2011 at 9:33 AM, William Stein <wst...@gmail.com> wrote:
>

> Hi Jon,
>
> You can now try using your algorithm to much higher precision, since I
> was able to compute the aplists to 50000 on two curves!!
>
> N = -38*a + 26
>   http://wstein.org/home/wstein/reu/2011/aplist_1756_50020.sobj
>
>
> N = -40*a + 24
>   http://wstein.org/home/wstein/reu/2011/aplist_1984_49800.sobj
>
>

Trying c4 = 1584*a + 1312 ; c6 =  110592*a + 62144  ; D = 16384*a + 9216
Elliptic Curve defined by y^2 = x^3 + x^2 + (-33*a-27)*x + (-139*a-81)

over Number Field in a with defining polynomial x^2 - x - 1

> William

[William Stein]

On Wed, Aug 10, 2011 at 11:11 PM, William Stein <wst...@gmail.com> wrote:

> Hi,
>
> To get to the first rank 2 we just need to deal with this one:
>
>
>   1476 -6*a + 42
> ['?',4,'?',0,2,-4,6,-8,-2,-8,-8,'?','?',-10,-14,6,8,-8,12,8,8,16,6,-14]
>
> I updated the aplist file for the above curve:
>
>     http://wstein.org/home/wstein/reu/2011/aplist-1476.sobj

I've updated that file again so it goes to norm conductor 50000.

William

>
> It now includes all a_P with Norm(P) < 42491.
>
>  -- William
>

--

[Paul Sharaba]

Thank you. Couldn't see it the first time.

[William Stein]

Hi,

What is the current status? Do we have every single curve up to norm
conductor 2000 except this heathen:

1476 -6*a + 42
['?',4,'?',0,2,-4,6,-8,-2,-8,-8,'?','?',-10,-14,6,8,-8,12,8,8,16,6,-14]

-- William

[Paul Sharaba]

We are missing that one and the following:

1856 -8*a + 48 ['?',-1,4,6,1,-5,8,'?','?',-7,-4,1,0,1,12,-4,8,6,5,8,10,1,-17,-8] 

[William Stein]

On Thu, Aug 11, 2011 at 11:39 AM, Paul Sharaba <paul.s...@gmail.com> wrote:
> We are missing that one and the following:
>
> 1856 -8*a + 48
> ['?',-1,4,6,1,-5,8,'?','?',-7,-4,1,0,1,12,-4,8,6,5,8,10,1,-17,-8]

I've updated the aplist file to beyond conductor norm 50000:

http://wstein.org/home/wstein/reu/2011/aplist-1856a.sobj

The sign in the functional equation is +1.

If Jon's still around, maybe he can try his program again with
enhanced precision.
If not, this is a good chance for somebody to try to learn to use Jon's program.

[Jonathan Bober]

No time for me to try right now. I'm at the airport.

But I think it will work on this curve. It looked close yesterday. I'm more worried about 1476. Something looked wrong. Maybe it is twisting by characters of really large conductor, or maybe there is some bug. Or maybe we finally found a case where the multiples of the periods are rational multiplesv instead of integer multiples.

[William Stein]

On Thu, Aug 11, 2011 at 12:07 PM, Jonathan Bober <jwb...@gmail.com> wrote:
> No time for me to try right now. I'm at the airport.
>
> But I think it will work on this curve. It looked close yesterday. I'm more
> worried about 1476. Something looked wrong. Maybe it is twisting by
> characters of really large conductor, or maybe there is some bug. Or maybe
> we finally found a case where the multiples of the periods are rational
> multiplesv instead of integer multiples.

Let E = curve we want to find of norm conductor 1476 of E is N =
2*3*(a-7) = p1 * p2 * p3, so the discriminant is p1^n1 * p2^n2 * p3^n3
for positive integers n1, n2, n3.

It turns out that there is a mod 7 congruence between the form of
level p1*p2*p3 and another form of level p2*p3. The congruent form
at level p2*p3 corresponds to the elliptic curve [0,a-1,a+1,-2*a,0] of
norm conductor 369 = 1476/4. The fact that there is such a
congruence I think implies that 7 divides n1; also a power of 2 can
divide n1. There are only mod 2 congruences with forms of level p1*p3
and p1*p2, so n2 and n3 are powers of 2. Jon -- maybe you can use
that the discriminant is of the form p1^(2^?*7^?) * p2^(2^?) *
p3^(2^?) to reduce the search space?

Another very interesting fact is that if we let F be the curve
[0,a-1,a+1,-2*a,0] of norm conductor 369, then E[7] = F[7]. This
means that if we consider the moduli space X_F(7) of all elliptic
curves with the same mod-7 torsion as F, then E corresponds to a point
on that moduli space. It turns out that Tom Fisher [0] wrote a huge
paper [1] about understanding this (and similar) moduli spaces.

[0] https://picasaweb.google.com/115360165819500279592/20110506BSDEngland#5605297295103255330

[1] http://www.dpmms.cam.ac.uk/~taf1000/papers/highercongr.html

Ben and I just worked out how to use [1] to try to find E. In theory
this should work, though there is no telling how hard it will be to
find the appropriate explicit point on the explicit quartic model for
X_F(7).
Here's the strategy (which Ben is working on):

1. Implement computation of the polynomials c_4(F) and c_6(F) from
page 14 of [1].
INPUT: F in K[x,y,z]
OUTPUT: c_4(F), c_6(F)
NOTE: On page 14 they are "a,b,c" but for this make them "x,y,z".

2. Type in the polynomials F, G on page 17 (theorem 3.6) of [1] and
d1, d2 on page 41 (theorem 6.10).
INPUT: c4, c6 in K
OUTPUT: Polynomials F, G, d1, d2 in K[x,y,z]. These depend on c4, c6.

3. Search for a solution to G=0 (or possibly one to F=0).

4. Use Theorem 6.8 on page 40 of [1] to write down the sought for
curve. It will be
y^2 = x^3 - 27*c_4(G)(point)/d2(point)^2 * X - 54 *
c6(G)(point)/d2(point)^3 [see paper for formula]

This is yet another alternative approach for finding the last
remaining curve of norm conductor <= 1831 (=first rank 2 curve) over
Q(sqrt(5)) that we don't know.

-- William