Cardinality of 512-bit curve

[sea21]

I would like to compute the cardinality of a 512-bit curve in Sage. For example, for y^2 = x^3 + a4.x + a6 mod p where

p=10566623376041669505825220895462627801380145726624712771836144280024219722297939525451022774579043146020265329009462778097121538072213487555318041328039599
a4=6557325753041215216697541661243177462316789827714644669153173743634615853451990973713919498189870050066222585898470391225455349541996119434962754618249308
a6=3887529832007272230349363633177495741990999092935596186265775681498612008751646149414901742385996951003068331698140100902826619262789957720578797201032523

When I tried to do so using the E.cardinality() function,

E = EllipticCurve(GF(p),[a4,a6])
#n = E.order()

I got the error similar to what was mentioned here: http://trac.sagemath.org/sage_trac/ticket/12403

When I tried to compute the order directly using GP with:

E = gp.ellinit([a4,a6],p)
ap = E.ellap(E,p)
n = p+1-ap

I got the following error:

*** at top-level..e[9] = ellinit(sage[7],sage[8])

***

*** overflow in t_INT-->long assignment

May I know how to work around this problem? Any help is greatly appreciated!

[Jeroen Demeyer]

On 05/15/2013 10:05 AM, sea21 wrote:
> E = gp.ellinit([a4,a6],p)
> ap = E.ellap(E,p)

You have the syntax wrong, it should be

sage: E = gp.ellinit([0,0,0,a4,a6])
sage: ap = E.ellap(p)

[sea21]

I see. Thanks for the correction! But when I typed in the right syntax, I then encountered the error:

*** at top-level: sage[9] = ellap(sage[7],sage[8]

***                                   ^----------------------

*** ellap: not enough modular polynomials

[Jeroen Demeyer]

On 05/15/2013 10:57 AM, sea21 wrote:
> I see. Thanks for the correction! But when I typed in the right syntax,
> I then encountered the error:
>
> *** at top-level: sage[9] = ellap(sage[7],sage[8]
> *** ^----------------------
> *** ellap: not enough modular polynomials

The PARI package in Sage only has the "small" database of modular
polynomials, suitable for primes up to about 350 bits. You would need
the full seadata database.

[John Cremona]

See http://pari.math.u-bordeaux.fr/packages.html. It should be easy
to download that and put it in the right place in the Sage
installation, which is SAGE_ROOT/local/shar/pari. There is already
the seadata in there but as Jeroen says, it is the small version.

John

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[John Cremona]

On 15 May 2013 10:47, John Cremona <john.c...@gmail.com> wrote:
> On 15 May 2013 10:25, Jeroen Demeyer <jdem...@cage.ugent.be> wrote:
>> On 05/15/2013 10:57 AM, sea21 wrote:
>>>
>>> I see. Thanks for the correction! But when I typed in the right syntax,
>>> I then encountered the error:
>>>
>>> *** at top-level: sage[9] = ellap(sage[7],sage[8]
>>> *** ^----------------------
>>> *** ellap: not enough modular polynomials
>>
>>
>> The PARI package in Sage only has the "small" database of modular
>> polynomials, suitable for primes up to about 350 bits. You would need the
>> full seadata database.
>
> See http://pari.math.u-bordeaux.fr/packages.html. It should be easy
> to download that and put it in the right place in the Sage
> installation, which is SAGE_ROOT/local/shar/pari. There is already
> the seadata in there but as Jeroen says, it is the small version.

I tested this. AFter putting the large seadata files into that place the code

sage: p
10566623376041669505825220895462627801380145726624712771836144280024219722297939525451022774579043146020265329009462778097121538072213487555318041328039599
sage: a4
6557325753041215216697541661243177462316789827714644669153173743634615853451990973713919498189870050066222585898470391225455349541996119434962754618249308
sage: a6
3887529832007272230349363633177495741990999092935596186265775681498612008751646149414901742385996951003068331698140100902826619262789957720578797201032523

sage: E = gp.ellinit([0,0,0,a4,a6])

sage: sage: ap = E.ellap(p)

runs fine.

John

[sea21]

Thanks for all the helpful replies! I understand the problem now. How do I install the full seadata package if I am using the Windows version of Sage on Virtual Box?

[leif]

sea21 wrote:
> Thanks for all the helpful replies! I understand the problem now. How do
> I install the full seadata package if I am using the Windows version of
> Sage on Virtual Box?

I don't know what the (Linux distribution of the) ova ships with, and
depends on whether you at all have internet access from the virtual machine.

Just download
http://pari.math.u-bordeaux.fr/pub/pari/packages/seadata.tgz (probably
from Windows), copy the file into some temporary directory in your
virtual machine, start a shell and change to that directory, and there do

$ tar xzf seadata.tgz # extract the GNU-zip-compressed tar
$ cp data/seadata/* "$SAGE_ROOT"/local/share/pari/seadata/
$ rm -rf data/

where $SAGE_ROOT is what

sage: print SAGE_ROOT

gives, the "root" of your Sage installation.

(You can of course delete seadata.tgz afterwards as well.)


Would it be worth providing an optional Sage package for this, for
convenience? (It would also be slightly smaller than PARI's .tgz ...)


-leif

> On Wednesday, May 15, 2013 8:07:48 PM UTC+8, John Cremona wrote:
>
> On 15 May 2013 10:47, John Cremona <john.c...@gmail.com

> >> The PARI package in Sage only has the "small" database of modular
> >> polynomials, suitable for primes up to about 350 bits. You would
> need the
> >> full seadata database.
> >
> > See http://pari.math.u-bordeaux.fr/packages.html

> <http://pari.math.u-bordeaux.fr/packages.html>. It should be easy

> > to download that and put it in the right place in the Sage
> > installation, which is SAGE_ROOT/local/shar/pari. There is already
> > the seadata in there but as Jeroen says, it is the small version.
>
> I tested this. AFter putting the large seadata files into that
> place the code
>
> sage: p
> 10566623376041669505825220895462627801380145726624712771836144280024219722297939525451022774579043146020265329009462778097121538072213487555318041328039599
>
> sage: a4
> 6557325753041215216697541661243177462316789827714644669153173743634615853451990973713919498189870050066222585898470391225455349541996119434962754618249308
>
> sage: a6
> 3887529832007272230349363633177495741990999092935596186265775681498612008751646149414901742385996951003068331698140100902826619262789957720578797201032523
>
> sage: E = gp.ellinit([0,0,0,a4,a6])
> sage: sage: ap = E.ellap(p)
>
> runs fine.

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[John Cremona]

On 15 May 2013 18:45, leif <not.r...@online.de> wrote:
> sea21 wrote:
>>
>> Thanks for all the helpful replies! I understand the problem now. How do
>> I install the full seadata package if I am using the Windows version of
>> Sage on Virtual Box?
>
>
> I don't know what the (Linux distribution of the) ova ships with, and
> depends on whether you at all have internet access from the virtual machine.
>
> Just download http://pari.math.u-bordeaux.fr/pub/pari/packages/seadata.tgz
> (probably from Windows), copy the file into some temporary directory in your
> virtual machine, start a shell and change to that directory, and there do
>
> $ tar xzf seadata.tgz # extract the GNU-zip-compressed tar
> $ cp data/seadata/* "$SAGE_ROOT"/local/share/pari/seadata/
> $ rm -rf data/
>
> where $SAGE_ROOT is what
>
> sage: print SAGE_ROOT
>
> gives, the "root" of your Sage installation.
>
> (You can of course delete seadata.tgz afterwards as well.)
>
>
> Would it be worth providing an optional Sage package for this, for
> convenience? (It would also be slightly smaller than PARI's .tgz ...)

I think it would. It would be a very simple spkg, but preferably we
should also try to catch situations when installing it would help the
user and prompt them to install it.

John

[sea21]

Thank you all for your prompt responses! It works fine for me now! =) It would definitely be useful to have an optional Sage package for this.

[John Cremona]

See http://trac.sagemath.org/sage_trac/ticket/14596 where all that is
needed is for someone to do the work!

John

[John Cremona]

Thanks to Jeroen for making this -- the spkg now exists and is
currently being spread around Sage's mirror sites. When that is done,
this will work:

sage: install_package("database_pari")


(assuming that you have write permission in the Sage directory where
files will be installed).

John