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 ?, ??, %magic, or help | ------------------------------------------------------- - ---------------------------------------------------------- -- Traceback (most recent call last): File "", 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 "", 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 ?, ??, %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: String Form: HasComplexMultiplication Namespace: Interactive Docstring: ( 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: String Form: 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: sage: E.has_cm?? Type: instancemethod Base Class: String Form: 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: String Form: 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 ?, ??, %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 ?, ??, %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: String Form: 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: 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 "", 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: