was@form:~$
was@form:~$
was@form:~$
was@form:~$ cd talks/
was@form:~/talks$ lt
total 1.4M
drwxr-xr-x   5 was was 1.3K 2006-02-27 11:30 2006-02-sa
ge-digipen
lrwxrwxrwx   1 was was   31 2006-02-15 12:46 2006-02-16
-bernoulli -> /home/was/papers/bernoulli/talk
drwxr-xr-x   3 was was  560 2006-02-04 08:57 2006-02-sa
ge_days
drwxr-xr-x   7 was was  552 2005-12-06 06:28 2005-12-05
-uw
drwxr-xr-x   7 was was  480 2005-12-03 02:54 2005-12-01
-ucla
drwxr-xr-x   3 was was  232 2005-11-30 14:51 2005-11-30
-avg_ranks
lrwxrwxrwx   1 was was   30 2005-11-27 00:46 texas-vish
igher -> ../papers/vishigher/texas-talk
was@form:~/talks$ cd 2006-02-sage-digipen/
lwas@form:~/talks/2006-02-sage-digipen$ ls
37a-lser.eps   current.toc
37a-lser.png   current.vrb
389A-lser.eps  deleted
389A-lser.png  E.sobj
A.sobj         guido.eps
auto           guido.jpg
car.eps        icon-5.eps
car.jpg        icon-5.jpg
cohen.eps      icon-5.png
cohen.jpg      L389a.eps
current.aux    L389a.png
current.dvi    log_during_talk.txt
current.log    log_during_talk-window2.txt
current.nav    pics
current.out    t
current.pdf    tire_pile.eps
current.ps     tire_pile.jpg
current.snm    website.eps
current.tex    website.png
was@form:~/talks/2006-02-sage-digipen$ o current.ps
ext =  ps
Opening current.ps using gv --swap --watch --quiet
gv --swap --watch --quiet "current.ps" 2>/dev/null 1>/d
ev/null&
was@form:~/talks/2006-02-sage-digipen$ sae
bash: sae: command not found
was@form:~/talks/2006-02-sage-digipen$ sage
-------------------------------------------------------
-
| SAGE Version 1.0.6, Build Date: 2006-02-26-0755
|
| Distributed under the GNU General Public License V2
|
| For help type <object>?, <object>??, %magic, or help
|
-------------------------------------------------------
-

----------------------------------------------------------
--
Traceback (most recent call last):
  File "<string>", line 1, in ?
  File "/home/was/s/local/lib/python2.4/site-packages/sage
/misc/preparser_ipython.py", line 8, in ?
    import sage.misc.interpreter
  File "/home/was/s/local/lib/python2.4/site-packages/sage
/misc/interpreter.py", line 82, in ?
    """
KeyboardInterrupt

WARNING: Failure executing code: 'import sage.misc.prepars
er_ipython;  sage.misc.preparser_ipython.magma_colon_equal
s=True'
----------------------------------------------------------
--
Traceback (most recent call last):
  File "<ipython console>", line 1, in ?
  File "/home/was/s/local/lib/python2.4/site-packages/sage
/all.py", line 34, in ?
    from sage.misc.all       import *
  File "/home/was/s/local/lib/python2.4/site-packages/sage
/misc/all.py", line 26, in ?
    from functional import *
  File "/home/was/s/local/lib/python2.4/site-packages/sage
/misc/functional.py", line 26, in ?
    from sage.rings.all import (RealField, ComplexField,
  File "/home/was/s/local/lib/python2.4/site-packages/sage
/rings/__init__.py", line 0, in ?
  File "/home/was/s/local/lib/python2.4/site-packages/sage
/interfaces/get_sigs.py", line 8, in my_sigint
    raise KeyboardInterrupt
KeyboardInterrupt

sage: ----------------------------------------------------
--------
Traceback (most recent call last):
  File "/home/was/s/local/lib/python2.4/site-packages/IPyt
hon/iplib.py", line 1757, in raw_input
    line = raw_input_original(prompt)
  File "/home/was/s/local/lib/python2.4/site-packages/sage
/interfaces/get_sigs.py", line 8, in my_sigint
    raise KeyboardInterrupt
KeyboardInterrupt

sage: ----------------------------------------------------
--------
Traceback (most recent call last):
  File "/home/was/s/local/lib/python2.4/site-packages/IPyt
hon/iplib.py", line 1757, in raw_input
    line = raw_input_original(prompt)
  File "/home/was/s/local/lib/python2.4/site-packages/sage
/interfaces/get_sigs.py", line 8, in my_sigint
    raise KeyboardInterrupt
KeyboardInterrupt

sage: ----------------------------------------------------
--------
Traceback (most recent call last):
  File "/home/was/s/local/lib/python2.4/site-packages/IPyt
hon/iplib.py", line 1757, in raw_input
    line = raw_input_original(prompt)
  File "/home/was/s/local/lib/python2.4/site-packages/sage
/interfaces/get_sigs.py", line 8, in my_sigint
    raise KeyboardInterrupt
KeyboardInterrupt

sage: clear
sage:
was@form:~/talks/2006-02-sage-digipen$ cd
was@form:~$ sage
--------------------------------------------------
| SAGE Version 1.0.6, Build Date: 2006-02-26-0755
| Distributed under the GNU General Public License
| For help type <object>?, <object>??, %magic, or
--------------------------------------------------

sage: !gap

            #########           ######         #####\
######           ###
         #############          ######         #####\
#######         ####
        ##############         ########        #####\
########       #####
       ###############         ########        #####\
   ######      #####
      ######         #         #########       #####\
    #####     ######
     ######                   ##########       #####\
    #####    #######
     #####                    ##### ####       #####\
   ######   ########
     ####                    #####  #####      #####\
########   ###  ####
     #####     #######       ####    ####      #####\
######    ####  ####
     #####     #######      #####    #####     #####\
#        ####   ####
     #####     #######      #####    #####     #####\
         #############
      #####      #####     ################    #####\
         #############
      ######     #####     ################    #####\
         #############
      ################    ##################   #####\
                ####
       ###############    #####        #####   #####\
                ####
         #############    #####        #####   #####\
                ####
          #########      #####          #####  #####\
                ####
                                                    \

     Information at:  http://www.gap-system.org
     Try '?help' for help. See also  '?copyright' an\
d  '?authors'

   Loading the library. Please be patient, this may \
take a while.
GAP4, Version: 4.4.6 of 02-Sep-2005, i686-pc-linux-g\
nu-gcc
Components:  small 2.1, small2 2.0, small3 2.0,
             small4 1.0, small5 1.0, small6 1.0,
             small7 1.0, small8 1.0, small9 1.0,
             small10 0.2, id2 3.0, id3 2.1, id4 1.0,
             id5 1.0, id6 1.0, id9 1.0, id10 0.1,
             trans 1.0, prim 2.1  loaded.
Packages:    FactInt 1.4.10, GAPDoc 0.9999,
             LAGUNA 3.3.1, TomLib 1.1.2  loaded.
gap>
sage: !gp
Reading GPRC: /etc/gprc ...Done.

      GP/PARI CALCULATOR Version 2.2.11 (alpha)
i686 running linux (ix86/GMP-4.1.4 kernel) 32-bit vers
ion
compiled: Jan 28 2006, gcc-4.0.2 20050808 (prerelease)
 (Ubuntu 4.0.1-4ubuntu9)
 (readline v5.0 enabled, extended help not available)

        Copyright (C) 2000-2005 The PARI Group

PARI/GP is free software, covered by the GNU
General Public License, and comes WITHOUT ANY
WARRANTY WHATSOEVER.

Type ? for help, \q to quit.
Type ?12 for how to get moral (and possibly technical)
 support.

parisize = 4000000, primelimit = 500000
?
Goodbye!
sage: !singular
                     SINGULAR
    /
 A Computer Algebra System for Polynomial Computations
   /   version 3-0-1

 0<
     by: G.-M. Greuel, G. Pfister, H. Schoenemann
   \   October 2005
FB Mathematik der Universitaet, D-67653 Kaiserslautern
    \
// ** executing /home/was/s/local/lib/Singular/3-0-1/L
IB/.singularrc
> Auf Wiedersehen.
sage: !maxima
  i i i i i i i       ooooo    o        ooooooo   oooo
o   ooooo
  I I I I I I I      8     8   8           8     8
 o  8    8
  I  \ `+' /  I      8         8           8     8
    8    8
   \  `-+-'  /       8         8           8      oooo
o   8oooo
    `-__|__-'        8         8           8
 8  8
        |            8     o   8           8     o
 8  8
  ------+------       ooooo    8oooooo  ooo8ooo   oooo
o   8

Copyright (c) Bruno Haible, Michael Stoll 1992, 1993
Copyright (c) Bruno Haible, Marcus Daniels 1994-1997
Copyright (c) Bruno Haible, Pierpaolo Bernardi, Sam St
eingold 1998
Copyright (c) Bruno Haible, Sam Steingold 1999-2003
------------------------------------------------------
--------

Maxima 5.9.2 http://maxima.sourceforge.net
Using Lisp CLISP 2.37 (2006-01-02)
Distributed under the GNU Public License. See the file
 COPYING.
Dedicated to the memory of William Schelter.
This is a development version of Maxima. The function
bug_report()
provides bug reporting information.
(%i1)
sage: E = EllipticCurve([1,2,3,4,5])
sage: E
 _6 = Elliptic Curve defined by y^2 + x*y + 3*y = x^3
+ 2*x^2 + 4*x + 5 over Rational Field
sage: E.
Display all 205 possibilities? (y or n)
E.CPS_height_bound
E.Hom
E.L1_vanishes
E.L_ratio
E.Lambda
E.Lseries
E.Lseries_at1
E.Lseries_deriv_at1
E.Lseries_extended
E.Np
E._AlgebraicScheme__A
E._AlgebraicScheme__divisor_groups
E._AlgebraicScheme_subscheme__G
E._EllipticCurve_generic__ainvs
E._EllipticCurve_generic__b_invariants
E._EllipticCurve_generic__base_ring
E._EllipticCurve_generic__discriminant
sage: E.Np(next_prime(2006))
 _7 = 2004
sage: E.Np(next_prime(10^10))
 _8 = 9999851383
sage: E.Np(next_prime(10^20))
 _9 = 100000000011093199520
sage: E.Np(next_prime(10^30))
_10 = 999999999999998209821055392921
sage: E.Np(next_prime(10^40))
_11 = 9999999999999999999925807069253028132026
sage: E.Np(next_prime(10^50))
_12 = 999999999999999999999999828651765886409162342223
62
sage: E.sea(next_prime(10^50))
_13 = 999999999999999999999999828651765886409162342223
62
sage: E.
Display all 209 possibilities? (y or n)
E.CPS_height_bound
E.Hom
E.L1_vanishes
E.L_ratio
E.Lambda
E.Lseries
E.Lseries_at1
E.Lseries_deriv_at1
E.Lseries_extended
E.Np
E._AlgebraicScheme__A
E._AlgebraicScheme__divisor_groups
E._AlgebraicScheme_subscheme__G
E._EllipticCurve_generic__ainvs
E._EllipticCurve_generic__b_invariants
E._EllipticCurve_generic__base_ring
E._EllipticCurve_generic__discriminant
E._EllipticCurve_generic__is_over_RationalField
E._EllipticCurve_rational_field__check_padic_hypothese
s
E._EllipticCurve_rational_field__conductor_pari
E._EllipticCurve_rational_field__minimal_model
E._EllipticCurve_rational_field__np
E._EllipticCurve_rational_field__pari_curve
E._EllipticCurve_rational_field__pari_double_prec
E._EllipticCurve_rational_field__pari_mincurve
E._base_ring
E._check_satisfies_equations
E._db_name
E._defining_params_
E._error_bad_coords
E._gap_
E._gap_init_
E._gp_
E._gp_init_
E._homset_class
E._interface_
E._interface_is_cached_
E._kash_
E._kash_init_
E._latex_
E._macaulay2_
E._macaulay2_init_
E._magma_
E._magma_init_
E._maple_
E._maple_init_
E._mathematica_
E._mathematica_init_
E._maxima_
E._maxima_init_
E._octave_
E._octave_init_
E._pari_
E._pari_init_
E._point_class
E._point_morphism_class
E._repr_
E._repr_type
E._sage_
E._set_conductor
E._set_cremona_label
E._set_gens
E._set_modular_degree
E._set_rank
E._set_torsion_order
E._singular_
E._singular_init_
E._validate
E.a1
E.a2
E.a3
E.a4
E.a6
E.a_invariants
E.affine_patch
E.ainvs
E.ambient_space
E.an
E.analytic_rank
E.anlist
E.ap
E.aplist
E.arithmetic_genus
E.b_invariants
E.base_extend
E.base_field
E.base_morphism
E.base_ring
E.base_scheme
E.c4
E.c6
E.c_invariants
E.category
E.change_ring
E.complex_area
E.conductor
E.coordinate_ring
E.cremona_label
E.database_curve
E.db
E.defining_ideal
E.defining_polynomial
E.defining_polynomials
E.dimension
E.discriminant
E.division_polynomial
E.divisor
E.divisor_group
E.divisor_of_function
E.dump
E.dumps
E.eval_modular_form
E.exclude
E.formal_group
E.formal_inverse
E.formal_log
E.formal_mult
E.formal_n_isogeny
E.formal_sigma
E.formal_w
E.formal_x
E.formal_y
E.gens
E.gens_certain
E.genus
E.geometric_genus
E.has_cm
E.heegner_discriminants
E.heegner_discriminants_list
E.heegner_index
E.heegner_index_bound
E.heegner_point_height
E.hom
E.identity_morphism
E.intersection
E.irreducible_components
E.is_good
E.is_integral
E.is_isomorphic
E.is_minimal
E.is_on_curve
E.is_ordinary
E.is_semistable
E.is_supersingular
E.is_surjective
E.isogeny_class
E.j_invariant
E.kodaira_type
E.label
E.local_coordinates
E.minimal_model
E.modular_degree
E.modular_parametrization
E.mwrank
E.mwrank_curve
E.newform
E.ngens
E.non_surjective
E.omega
E.p_isogenous_curves
E.padic_E2
E.padic_height
E.padic_regulator
E.pari_curve
E.pari_mincurve
E.period_lattice
E.point
E.point_homset
E.point_search
E.point_set
E.q_eigenform
E.q_expansion
E.quadratic_twist
E.rank
E.rational_points
E.real_components
E.reduce
E.reducible_primes
E.regulator
E.rename
E.reset_name
E.root_number
E.satisfies_heegner_hypothesis
E.saturation
E.save
E.sea
E.selmer_rank_bound
E.sha_an
E.shabound
E.shabound_kato
E.shabound_kolyvagin
E.sigma
E.silverman_height_bound
E.simon_two_descent
E.structure_morphism
E.tamagawa_number
E.tamagawa_product
E.three_selmer_rank
E.torsion_order
E.torsion_polynomial
E.torsion_subgroup
E.two_descent
E.two_descent_simon
E.two_selmer_shabound
sage: E.simon_two_descent ()
_14 = (1, 1, [(63 : 864 : 1)])
sage: F = magma(E)
sage: F
_16 = Elliptic Curve defined by y^2 + x*y + 3*y = x^3
+ 2*x^2 + 4*x + 5 over Rational Field
sage: F.
Display all 197 possibilities? (y or n)
F.AbelianGroup
F.AnalyticRank
F.AnalyticRankNumberOfTerms
F.AnalyticRankQuadraticTwist
F.Automorphism
F.BadPlaces
F.BadPrimes
F.Coefficients
F.Conductor
F.ConjecturalRegulator
F.CremonaReference
F.Discriminant
F.DivisionFunction
F.DivisionPolynomial
F.DivisionPsi
F.ElementToSequence
F.EllipticExponential
F.EllipticLogarithm
F.Eltseq
F.EulerFactor
F.FormalLog
F.FrobeniusMap
F.FrobeniusTraceDirect
F.FunctionField
F.Generators
F.HasComplexMultiplication
F.HeegnerPoint
F.HeightPairingLattice
F.HeightPairingMatrix
F.Hom
F.HyperellipticCurve
F.HyperellipticPolynomials
F.IdentityIsogeny
F.IdentityMap
sage: F.HasComplexMultiplication?
Type:           MagmaFunctionElement
Base Class:     <class 'sage.interfaces.magma.MagmaFun
ctionElement'>
String Form:    HasComplexMultiplication
Namespace:      Interactive
Docstring:
    (<CrvEll> E) -> BoolElt, RngIntElt

    Determines whether E (defined over Q or a number f
ield) has CM or not. If it does, also returns the disc
riminant of the quadratic order of CM.



sage: F.HasComplexMultiplication()
_18 = false
sage: E.has_cm ()
_19 = False
sage: E.has_cm?
Type:           instancemethod
Base Class:     <type 'instancemethod'>
String Form:    <bound method EllipticCurve_rational_f
ield.has_cm of Elliptic Curve defined by y^2 + x*y + 3
*y = x^3 + 2*x^2 + 4*x + 5 over Rational Field>
Namespace:      Interactive
File:           /home/was/s/local/lib/python2.4/site-p
ackages/sage/schemes/elliptic_curves/ell_rational_fiel
d.py
Definition:     E.has_cm(self)
Docstring:
    <no docstring>

sage: E.has_cm??
Type:           instancemethod
Base Class:     <type 'instancemethod'>
String Form:    <bound method EllipticCurve_rational_f
ield.has_cm of Elliptic Curve defined by y^2 + x*y + 3                                     *y = x^3 + 2*x^2 + 4*x + 5 over Rational Field>
Namespace:      Interactive
File:           /home/was/s/local/lib/python2.4/site-p                                     ackages/sage/schemes/elliptic_curves/ell_rational_fiel
d.py
Definition:     E.has_cm(self)
Source:
def has_cm(self):
        return self.j_invariant() in [0, 54000, -12288
000, 1728, \
                                      287496, -3375, 1
6581375, 8000, \
                                      -32768,  -884736
, -884736000,\
                                      -147197952000, -
262537412640768000]
sage: E.has_cm??
Type:           instancemethod
Base Class:     <type 'instancemethod'>
String Form:    <bound method EllipticCurve_rational_field.has_cm
^3 + 2*x^2 + 4*x + 5 over Rational Field>
Namespace:      Interactive
File:           /home/was/s/local/lib/python2.4/site-packages/sag
Definition:     E.has_cm(self)
Source:
def has_cm(self):
        return self.j_invariant() in [0, 54000, -12288000, 1728,
                                      287496, -3375, 16581375, 80
                                      -32768,  -884736, -88473600
                                      -147197952000, -26253741264
sage: m = Magma()
sage: x = m(100)
sage: x
_25 = 100
sage:
Exiting SAGE (CPU time 0m0.71s, Wall time 10m24.42s).
was@form:~$ sage
--------------------------------------------------------
| SAGE Version 1.0.6, Build Date: 2006-02-26-0755      |
| Distributed under the GNU General Public License V2  |
| For help type <object>?, <object>??, %magic, or help |
--------------------------------------------------------

sage: E = EllipticCurve([1,2,3,4,5])
sage: E
 _2 = Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4
*x + 5 over Rational Field
sage: E._repr_()
 _3 = 'Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 +
4*x + 5 over Rational Field'
sage: E.rename('Joe curve')
sage: E
 _5 = Joe curve
sage: E.rename('E')
sage: E
 _7 = E
sage: E.save('a.sobj')
sage:
Exiting SAGE (CPU time 0m0.04s, Wall time 4m5.48s).
was@form:~$ sage
--------------------------------------------------------
| SAGE Version 1.0.6, Build Date: 2006-02-26-0755      |
| Distributed under the GNU General Public License
| For help type <object>?, <object>??, %magic, o
------------------------------------------------

sage: E = load('a.sobj')
sage: E
 _2 = E
sage: latex(E)
 _3 = 'y^2 + xy + 3y = x^3 + 2x^2 + 4x + 5 '
sage: k = GF(9)
sage: k
 _5 = Finite Field in a of size 3^2
sage: M = MatrixSpace(k, 10)
sage: M
 _7 = Full MatrixSpace of 10 by 10 dense matrice
ld in a of size 3^2
sage: A = M.random_element()
sage: A
 _9 =
[2 1 2 1 1 1 1 1 1 1]
[2 1 1 2 2 2 2 2 2 2]
[1 1 1 2 2 1 2 1 1 1]
[2 1 2 2 1 1 1 2 2 2]
[1 2 2 2 1 2 2 1 2 2]
[1 2 1 1 2 2 2 1 1 2]
[2 1 1 2 1 2 1 2 1 2]
[2 2 1 1 2 2 1 1 1 1]
[2 1 1 2 2 2 1 2 1 1]
[2 1 2 1 1 2 1 2 2 2]
sage: M
_10 = Full MatrixSpace of 10 by 10 dense matrice
s over Finite Field in a of size 3^2
sage: A = M([k.random_element() for _ in range(1
00)]
   ....:
   ....: )
sage: A
_12 =
[      2       0       2   a + 2 2*a + 1       a
     2*a 2*a + 2 2*a + 2   a + 1]
[      0   a + 2       1     2*a       0       0
   a + 2       0     2*a 2*a + 1]
[2*a + 1       1       1   a + 2 2*a + 2       2
       0 2*a + 1       2   a + 1]
[      0 2*a + 2       0 2*a + 1   a + 1       1
       1       1       1       0]
[  a + 2   a + 1 2*a + 2 2*a + 2 2*a + 1     2*a
       a 2*a + 2       0       a]
[  a + 1       a       2       2     2*a   a + 2
       1   a + 2     2*a       2]
[      0 2*a + 2       1       1       1   a + 1
 2*a + 2       1   a + 2       1]
[  a + 2       0     2*a     2*a   a + 1 2*a + 1
 2*a + 2     2*a       2     2*a]
[    2*a       0       1     2*a 2*a + 1       a
     2*a       2   a + 2 2*a + 1]
[      0       a       2 2*a + 1       a       a
       a 2*a + 1 2*a + 2       2]
sage: A
_13 =
[      2       0       2   a + 2 2*a + 1       a     2*a 2*a + 2 2*a + 2   a + 1]
[      0   a + 2       1     2*a       0       0   a + 2       0     2*a 2*a + 1]
[2*a + 1       1       1   a + 2 2*a + 2       2       0 2*a + 1       2   a + 1]
[      0 2*a + 2       0 2*a + 1   a + 1       1       1       1       1       0]
[  a + 2   a + 1 2*a + 2 2*a + 2 2*a + 1     2*a       a 2*a + 2       0       a]
[  a + 1       a       2       2     2*a   a + 2       1   a + 2     2*a       2]
[      0 2*a + 2       1       1       1   a + 1 2*a + 2       1   a + 2       1]
[  a + 2       0     2*a     2*a   a + 1 2*a + 1 2*a + 2     2*a       2     2*a]
[    2*a       0       1     2*a 2*a + 1       a     2*a       2   a + 2 2*a + 1]
[      0       a       2 2*a + 1       a       a       a 2*a + 1 2*a + 2       2]
sage: print latex(A)
\left(\begin{array}{rrrrrrrrrr}
2&0&2&a + 2&2a + 1&a&2a&2a + 2&2a + 2&a + 1\\
0&a + 2&1&2a&0&0&a + 2&0&2a&2a + 1\\
2a + 1&1&1&a + 2&2a + 2&2&0&2a + 1&2&a + 1\\
0&2a + 2&0&2a + 1&a + 1&1&1&1&1&0\\
a + 2&a + 1&2a + 2&2a + 2&2a + 1&2a&a&2a + 2&0&a\\
a + 1&a&2&2&2a&a + 2&1&a + 2&2a&2\\
0&2a + 2&1&1&1&a + 1&2a + 2&1&a + 2&1\\
a + 2&0&2a&2a&a + 1&2a + 1&2a + 2&2a&2&2a\\
2a&0&1&2a&2a + 1&a&2a&2&a + 2&2a + 1\\
0&a&2&2a + 1&a&a&a&2a + 1&2a + 2&2
\end{array}\right)
sage: A._latex_?
Type:           instancemethod
Base Class:     <type 'instancemethod'>
String Form:
<bound method Matrix_generic_dense_field._latex_ of [      2       0       2   a +
2 2*a + 1      <...> a + 2 2*a + 1]
            [      0       a       2 2*a + 1       a       a
2       2]>
Namespace:      Interactive
File:           /home/was/s/local/lib/python2.4/site-pack
Definition:     A._latex_(self)
Docstring:
    <no docstring>

sage: A._latex_??
sage: log_dvi()
_17 = DVI Logger
sage: log_html()
_18 = HTML Logger
sage: E.q_expansion(20)
_19 = q + q^2 - q^4 - 3*q^5 - q^7 - 3*q
^16 + 5*q^17 - 3*q^18 + 4*q^19 + O(q^20
sage: E.q_expansion(20)
_20 = q + q^2 - q^4 - 3*q^5 - q^7 - 3*q
^16 + 5*q^17 - 3*q^18 + 4*q^19 + O(q^20
sage: F = magma(E)
sage: F
_22 = Elliptic Curve defined by y^2 + x
*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over R
ational Field
sage: latex(F)
_23 = '\\begin{verbatim}Elliptic Curve
defined by y^2 + x*y + 3*y = x^3 + 2*x^
2 + 4*x + 5 over Rational Field\\end{ve
rbatim}'
sage: f = maxima('sin(x)*x + exp(x)')
sage: f
_25 = x*sin(x) + %e^x
sage: latex(f)
_26 = 'x\\sin{}x + e^{x}'
sage: a = mathematica('Sin[40]')
sage: a
_28 = Sin[40]
sage: float(a)
_29 = 0.74511316047934883
sage: a = factor(93408234802834)
sage: a
_31 = 2 * 7 * 479287 * 13920713
sage: factor?
sage: factor(a, algorithm='kash')
---------------------------------------
---------------------
Traceback (most recent call last):
  File "<ipython console>", line 1, in
?
  File "/home/was/s/local/lib/python2.4
/site-packages/sage/rings/arith.py", li
ne 997, in factor
    raise TypeError, "unable to factor
%s"%n
TypeError: unable to factor 2 * 7 * 479
287 * 13920713

sage: a
_34 = 2 * 7 * 479287 * 13920713
sage: factor(-2006, algorithm='kash')
_35 = -1 * 2 * 17 * 59
sage: m = Mwrank()
sage: m('[1,2,3,4,5]')
_37 = '\nCurve [1,2,3,4,5] :\tWorking w
ith minimal curve [1,-1,0,4,3]\n\t[u,r,
s,t] = [1,-1,0,-1]\nBasic pair: I=-183,
 J=-6858\ndisc=-71546112\n2-adic index
bound = 2\nBy Lemma 5.1(b), 2-adic inde
x = 1\n2-adic index = 1\nOne (I,J) pair
\n*** BSD give two (I,J) pairs\nLooking
 for quartics with I = -183, J = -6858\
nLooking for Type 3 quartics:\nTrying p
ositive a from 1 up to 6 (square a firs
t...)\n(1,2,-9,22,-11)\t--nontrivial...
(x:y:z) = (1 : 1 : 0)\nPoint = [2:3:1]\
n\theight = 1.26539379176449\nRank of B
=im(eps) increases to 1\n(4,3,6,3,-4)\t
--trivial\nTrying positive a from 1 up
to 6 (...then non-square a)\nTrying neg
ative a from -1 down to -5\n(-4,3,6,3,4
)\t--trivial\nFinished looking for Type
 3 quartics.\nMordell rank contribution
 from B=im(eps) = 1\nSelmer  rank contr
ibution from B=im(eps) = 1\nSha     ran
k contribution from B=im(eps) = 0\nMord
ell rank contribution from A=ker(eps) =
 0\nSelmer  rank contribution from A=ke
r(eps) = 0\nSha     rank contribution f
rom A=ker(eps) = 0\nRank = 1\n\nRegulat
or (before saturation) = 1.265393791764
49\nSearching for points (bound = 10)..
.done\nRegulator (after searching) = 1.
26539379176449\nSaturating (bound = 100
)...finished saturation (index was 1)\n
Regulator (after saturation) = 1.265393
79176449\nTransferring points back to o
riginal curve [1,2,3,4,5]\n\nGenerator
1 is [1:2:1]; height 1.26539379176449\n
\nRegulator = 1.26539379176449\n\nThe r
ank and full Mordell-Weil basis have be
en determined unconditionally.\n (0.196
 seconds)'
sage: print m('[1,2,3,4,5]')

Curve [1,2,3,4,5] :     Working with mi
nimal curve [1,-1,0,4,3]
        [u,r,s,t] = [1,-1,0,-1]
Basic pair: I=-183, J=-6858
disc=-71546112
2-adic index bound = 2
By Lemma 5.1(b), 2-adic index = 1
2-adic index = 1
One (I,J) pair
*** BSD give two (I,J) pairs
Looking for quartics with I = -183, J =
 -6858
Looking for Type 3 quartics:
Trying positive a from 1 up to 6 (squar
e a first...)
(1,2,-9,22,-11) --nontrivial...(x:y:z)
= (1 : 1 : 0)
Point = [2:3:1]
        height = 1.26539379176449
Rank of B=im(eps) increases to 1
(4,3,6,3,-4)    --trivial
Trying positive a from 1 up to 6 (...th
en non-square a)
Trying negative a from -1 down to -5
(-4,3,6,3,4)    --trivial
Finished looking for Type 3 quartics.
Mordell rank contribution from B=im(eps
) = 1
Selmer  rank contribution from B=im(eps
) = 1
Sha     rank contribution from B=im(eps
) = 0
Mordell rank contribution from A=ker(ep
s) = 0
Selmer  rank contribution from A=ker(ep
s) = 0
Sha     rank contribution from A=ker(ep
s) = 0
Rank = 1

Regulator (before saturation) = 1.26539
379176449
Searching for points (bound = 10)...don
e
Regulator (after searching) = 1.2653937
9176449
Saturating (bound = 100)...finished sat
uration (index was 1)
Regulator (after saturation) = 1.265393
79176449
Transferring points back to original cu
rve [1,2,3,4,5]

Generator 1 is [1:2:1]; height 1.265393
79176449

Regulator = 1.26539379176449

The rank and full Mordell-Weil basis ha
ve been determined unconditionally.
 (0.176 seconds)
sage: ASSERT: "!icon.isEmpty()" in /roo
t/3.5/kdebase/kdebase-3.5.0/./libkonq/k
onq_pixmapprovider.cc (79)
ASSERT: "!icon.isEmpty()" in /root/3.5/
kdebase/kdebase-3.5.0/./libkonq/konq_pi
xmapprovider.cc (79)
ASSERT: "!icon.isEmpty()" in /root/3.5/
kdebase/kdebase-3.5.0/./libkonq/konq_pi
xmapprovider.cc (79)
ASSERT: "!icon.isEmpty()" in /root/3.5/
kdebase/kdebase-3.5.0/./libkonq/konq_pi
xmapprovider.cc (79)

sage: n = magma(-2006)
sage: n.
Display all 729 possibilities? (y or n)
sage: n.
Display all 729 possibilities? (y or n)
n.'*'
n.'*:='
n.'+'
n.'+:='
n.'-'
n.'-:='
n.'/'
n.'/:='
n.'@'
n.'^'
n.'^:='
n.'div'
n.'div:='
n.'eq'
n.'in'
n.'le'
n.'lt'
n.'mod'
n.'mod:='
n.AGL
n.AGammaL
n.ASL
n.ASigmaL
n.Abs
n.AbsIrrApplyGalois
n.AbsoluteValue
n.AbsolutelyIrreducibleModulesDelete
n.AbsolutelyIrreducibleRepresentationsA
pply
n.AbsolutelyIrreducibleRepresentationsD
elete
n.AbsolutelyIrreducibleRepresentationsP
rocessDegree
n.AbsolutelyIrreducibleRepresentationsP
rocessGroup
n.Access
n.AdditiveCyclicCode
n.AdditiveMacWilliamsTransform
n.AdditiveQuasiCyclicCode
n.AffineGammaLinearGroup
n.AffineGeneralLinearGroup
n.AffineSigmaLinearGroup
n.AffineSpecialLinearGroup
n.Alarm
n.Alt
n.AlternatingCharacterTable
n.AlternatingGroup
n.AtkinModularEquation
n.AtkinModularPolynomial
n.AutPSp
n.BBSModulus
n.BSGSProcess
n.BaerDerivation
n.BaseMPolynomial
n.Bernoulli
n.BernoulliApproximation
n.BernoulliNumber
n.BernoulliPolynomial
n.BesselFunction
n.BinaryQuadraticForms
n.Binomial
n.BipartiteGraph
n.BlackboxGroup
n.BlumBlumShub
n.BlumBlumShubModulus
n.BorderedDoublyCirculantQRCode
n.BraidGroup
n.BrandtModule
n.BrandtModuleDimension
n.CalabiYau
n.CambridgeMatrix
n.CanIdentifyGroup
n.CanonicalModularEquation
n.CanonicalModularPolynomial
n.CarmichaelLambda
n.Catalan
n.Character
n.ChebyshevFirst
n.ChebyshevSecond
n.ChebyshevT
n.ChebyshevU
n.ClassGroupGenerationBound
n.ClassNumber
n.ClassNumberApproximationBound
n.ClassTwo
n.ClassicalModularEquation
n.ClassicalModularPolynomial
n.ClassicalMultiplication
n.CodeToString
n.CompleteDigraph
n.CompleteGraph
n.ComplexConjugate
n.ComplexField
n.CongruenceSubgroup
n.ConjugacyClassesGL
n.Conjugate
n.Conjugates
n.ConstaCyclicCode
n.ConwayPolynomial
n.CordaroWagnerCode
n.CountPGroups
n.Covariant
n.CreateFanoData
n.CreateK3Data
n.Cunningham
n.Cusp
n.CyclicCode
n.CyclicGroup
n.CyclotomicField
n.CyclotomicPolynomial
n.DecimalToBitPrecision
n.DefinitionSets
n.DegeneracyCosetRepsInner
n.DelsarteGoethalsCode
n.DicksonFirst
n.DicksonSecond
n.DifferenceSet
n.DihedralGroup
n.DimensionCuspFormsGamma0
n.DimensionCuspFormsGamma1
n.DimensionNewCuspFormsGamma0
n.DimensionNewCuspFormsGamma1
n.DirichletGroup
n.DirichletGroupNew
n.DivisorSigma
n.Divisors
n.DoublyCirculantQRCode
n.DoublyCirculantQRCodeGF4
n.ECM
n.ECMFactoredOrder
n.ECMOrder
n.ECMSteps
n.EffectivePossibilities
n.EffectiveSubcanonicalCurves
n.Eisenstein
n.ElementToSequence
n.ElementaryToHomogeneousMatrix
n.ElementaryToMonomialMatrix
n.ElementaryToPowerSumMatrix
n.ElementaryToSchurMatrix
n.Eltseq
n.EmptyDigraph
n.EmptyGraph
n.EmptyMultiDigraph
n.EmptyMultiGraph
n.EmptyNetwork
n.EuclideanNorm
n.EulerPhi
n.EulerPhiInverse
n.EulerianNumber
n.EvenWeightCode
n.ExistsConwayPolynomial
n.ExtendedGreatestCommonDivisor
n.ExtensionsOfElementaryAbelianGroup
n.ExtraSpecialGroup
n.Facint
n.FactoredCarmichaelLambda
n.FactoredEulerPhi
n.FactoredEulerPhiInverse
n.Factorial
n.FactorialValuation
n.Factorisation
n.FactorisationToInteger
n.Factorization
n.FactorizationToInteger
n.Fano
n.FanoBaskets
n.FanoIsolatedBaskets
n.Fibonacci
n.FindN
n.FindPowerSeries
n.FindPowerSeriesForChabauty
n.FiniteAffinePlane
n.FiniteField
n.FiniteProjectivePlane
n.FreeAbelianGroup
n.FreeGroup
n.FreeMonoid
n.FreeNilpotentGroup
n.FreeSemigroup
n.FrobeniusMap
n.FullDirichletGroup
n.FundamentalDiscriminant
n.G2
n.G2Reduced
n.GCD
n.GCDSup
n.GF
n.GL
n.GL2IrreducibleSolubleSubgroups
n.GL2IrreducibleSubgroups
n.GL3IrreducibleSolubleSubgroups
n.GO
n.GOMinus
n.GOPlus
n.GR
n.GU
n.GaloisField
n.GaloisRing
n.Gamma0
n.Gamma1
n.GammaUpper0
n.GammaUpper1
n.GaussNumber
n.GaussianBinomial
n.GaussianFactorial
n.Gcd
n.GcdSup
n.GegenbauerPolynomial
n.GeneralLinearGroup
n.GeneralOrthogonalGroup
n.GeneralOrthogonalGroupMinus
n.GeneralOrthogonalGroupPlus
n.GeneralUnitaryGroup
n.GeneralizedFibonacciNumber
n.GenerateGraphs
n.GeneratepGroups
n.GoethalsCode
n.GoethalsDelsarteCode
n.GraphInBytes
n.GraphSizeInBytes
n.GreatestCommonDivisor
n.Groupsp7
n.HadamardMatrixFromInteger
n.HarmonicNumber
n.HeckeOperator
n.HeilbronnCremona
n.HeilbronnMerel
n.HermitePolynomial
n.HermitianCode
n.HermitianFunctionField
n.HilbertClassPolynomial
n.HilbertMatrix
n.HilbertPolynomialOfCurve
n.Hom
n.HomogeneousToElementaryMatrix
n.HomogeneousToMonomialMatrix
n.HomogeneousToPowerSumMatrix
n.HomogeneousToSchurMatrix
n.HyperbolicCoxeterGraph
n.HyperbolicCoxeterMatrix
n.HyperellipticCurveOfGenus
n.IharaBound
n.Ilog
n.Ilog2
n.ImprimitiveReflectionGroup
n.IneffectivePossibilities
n.IneffectiveSubcanonicalCurves
n.IntegerRing
n.IntegerToSequence
n.IntegerToString
n.Integers
n.InternalFactorizationTableInit
n.InternalGBLieAlgebra
n.InternalHardConjCountSet
n.InternalJacRelations
n.InternalQUEA
n.Internal_p4_list
n.Internal_p5_list
n.Internal_p6_list
n.Internal_p7_list
n.Intseq
n.InverseMod
n.Iroot
sage: n.Factorization?
sage: n.Factorization()
_41 = [ <2, 1>, <17, 1>, <59, 1> ]
sage: n.Factorization(nvals=2)
_42 = ([ <2, 1>, <17, 1>, <59, 1> ], -1                                                    )
sage: