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How to use modular polynomials in Sage?

asked 2018-02-26 14:41:17 -0500

ninho gravatar image

I want to perform some calculations using modular polynomials, and I saw that Sage already offers some databases for this. Most notably, I came across this. But, I couldn't find any documentation or tutorial as how this database is actually used, or what it actually includes. I can do something like this, but it doesn't give much information:

db = ClassicalModularPolynomialDatabase()
print db

Is there any tutorial or documentation as how to use modular polynomials in Sage?

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answered 2018-02-26 16:39:39 -0500

dan_fulea gravatar image

Here is a sample code, that recovers $\Phi_2$ and $\Phi_4$ from

https://math.mit.edu/~drew/ClassicalModPolys.html

using sage code. (Note that the polynomial displayed in the above link is $\Phi_2$, not $\Phi_4$ as stated.) One needs the kohel database. (In manjaro i installed it from the AUR, the name of the package was sage-data-kohel, referenced here, sage-data-kohel so i can not state here certainly that sage -i database_kohel does the job. A similar command, possibly with a suffix should install the database properly.)

Then

[dan@k7 ~]$ sage
┌────────────────────────────────────────────────────────────────────┐
│ SageMath version 8.1, Release Date: 2017-12-07                     │
│ Type "notebook()" for the browser-based notebook interface.        │
│ Type "help()" for help.                                            │
└────────────────────────────────────────────────────────────────────┘
sage: PHI = ClassicalModularPolynomialDatabase()
sage: var( 'X,Y' );
sage: PHI[2]
-j0^2*j1^2 + j0^3 + 1488*j0^2*j1 + 1488*j0*j1^2 + j1^3 - 162000*j0^2 + 40773375*j0*j1 - 162000*j1^2 + 8748000000*j0 + 8748000000*j1 - 157464000000000
sage: PHI[2](X,Y)
-X^2*Y^2 + X^3 + 1488*X^2*Y + 1488*X*Y^2 + Y^3 - 162000*X^2 + 40773375*X*Y - 162000*Y^2 + 8748000000*X + 8748000000*Y - 157464000000000

reproduces the (symmetrical) $\Phi_2(X,Y)$ polynomial from the above link. To have the data from the links

we can then ask for

PHI = ClassicalModularPolynomialDatabase()
for N in (2, 3, 4):
    print "N = %s" % N
    data = [ ( mono.degrees(), coeff ) for coeff, mono in PHI[N] ]
    data . sort()
    for degrees, coeff in data:
        # we show only the half of the coefficients of the symmetrical pol PHI[N]
        if degrees[0] >= degrees[1]:
            print list(degrees), coeff
    print

and the results are:

N = 2
[0, 0] -157464000000000
[1, 0] 8748000000
[1, 1] 40773375
[2, 0] -162000
[2, 1] 1488
[2, 2] -1
[3, 0] 1

N = 3
[1, 0] 1855425871872000000000
[1, 1] -770845966336000000
[2, 0] 452984832000000
[2, 1] 8900222976000
[2, 2] 2587918086
[3, 0] 36864000
[3, 1] -1069956
[3, 2] 2232
[3, 3] -1
[4, 0] 1

N = 4
[0, 0] 280949374722195372109640625000000000000
[1, 0] -364936327796757658404375000000000000
[1, 1] -94266583063223403127324218750000
[2, 0] 158010236947953767724187500000000
[2, 1] 188656639464998455284287109375
[2, 2] 26402314839969410496000000
[3, 0] -22805180351548032195000000000
[3, 1] 12519806366846423598750000
[3, 2] -914362550706103200000
[3, 3] 2729942049541120
[4, 0] 24125474716854750000
[4, 1] 1194227244109980000
[4, 2] 1425220456750080
[4, 3] 80967606480
[4, 4] 7440
[5, 0] -8507430000
[5, 1] 561444609
[5, 2] -2533680
[5, 3] 2976
[5, 4] -1
[6, 0] 1

The relevant py-file may be located as follows:

sage: M = sage.databases.db_modular_polynomials
sage: M.__file__
'/usr/lib/python2.7/site-packages/sage/databases/db_modular_polynomials.pyc'

and the py (not its compiled version, with suffix pyc) starts with:

"""
Database of Modular Polynomials
"""
#######################################################################
#       Copyright (C) 2006 William Stein <wstein@gmail.com>
#       Copyright (C) 2006 David Kohel <kohel@maths.usyd.edu.au>
#       Copyright (C) 2016 Vincent Delecroix <vincent.delecroix@labri.fr>
#
#  Distributed under the terms of the GNU General Public License (GPL)
#  as published by the Free Software Foundation; either version 2 of
#  the License, or (at your option) any later version.
#                  http://www.gnu.org/licenses/
#*****************************************************************************

from __future__ import print_function, absolute_import

def _dbz_to_string(name):

and so on. Hope, this is a good starting point to get the modular polynomials, the canonical equations for the modular curves in the family $X_0(N)$. (For me, PHI[19] was printed, PHI[20] was missing, LookupError: filename /usr/share/kohel/PolMod/Cls/pol.020.dbz does not exist.)

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Thanks for the beautiful answer.

ninho gravatar imageninho ( 2018-02-27 03:02:35 -0500 )edit

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Asked: 2018-02-26 14:41:17 -0500

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Last updated: Feb 26