SAGE is open source mathematics software which creates a viable free open source alternative to Magma, Maple, Mathematica, and Matlab.
General and Advanced Pure and Applied Mathematics
Use SAGE for studying a huge range of mathematics, including algebra, calculus, elementary to very advanced number theory, cryptography, numerical computation, commutative algebra, group theory, combinatorics, graph theory, and exact linear algebra.
Use an Open Source Alternative
By using SAGE you help to support a viable open source alternative to Magma, Maple, Mathematica, and MATLAB. SAGE includes many high-quality open source math packages.
Use Most Mathematics Software from Within SAGE
SAGE makes it easy for you to use most mathematics software together. SAGE includes interfaces to Magma, Maple, Mathematica, MATLAB, and MuPAD, and the free programs Axiom, GAP, GP/PARI, Macaulay2, Maxima, Octave, and Singular.
Use a Standard Programming Language
You work with SAGE using the highly regarded scripting language Python instead of an obscure language designed for a particular mathematics program. You can write programs that combine serious mathematics with anything else.
Version 4.8:
- #12308: Update Cremona's table of elliptic curves for Sage (R. Andrew Ohana)
- #1385: Re-organize number field element inheritance hierarchy
- #5377: weird output for trivial class group
- #8664: Upgrade Sage's MPIR spkg to version 2.1.3 (Mike Hansen, Leif Leonhardy)
- #9389: Bug in polynomials over residue fields (was: sage crashing when computing local_data() for an EC)
- #9858: FLINT 1.5.0.p5's test suite fails to build with new GMP/MPIR (Leif Leonhardy)
- #9992: Python scripts try to run before Python is built. (Leif Leonhardy)
- #10141: bug in printing trivial class groups (trivial to fix)
- #10155: Implementation of the Cyclic Sieving Phenomenon (Christian Stump)
- #10200: Catch Sphinx WARNING or ERROR
- #10275: matrix output strangeness for cached methods (John Palmieri, Julian Rueth)
- #10505: Round values returned by CPLEX when the variable's type is integer/binary (Nathann Cohen)
- #10730: simon_two_descent -- reports points as being independent, but they are not
- #10910: Avoid nfinit while factoring polynomials
- #10954: Expose weight options in distance method (Nathann Cohen)
- #11099: digraphs.RandomDirectedGNM (Nathann Cohen)
- #11119: ECL 11.1.1 fails on Cygwin
- #11367: Rounding values, and inconsistencies in graph methods (Nathann Cohen)
- #11560: update sympy to 0.7.1 (François Bissey)
- #11587: update Cremona's tables for Sage (R. Andrew Ohana)
- #11742: Implementation of class for cores (Anne Schilling, Mike Zabrocki)
- #11812: traceback with load and attach of .sage files (Marco Streng)
- #11846: Independent set through Linear Programming (sometimes, it is faster !) (Nathann Cohen)
- #11869: Horrible bug in number field conversion (Jeroen Demeyer)
- #11882: update readline to 6.2 (John Palmieri)
- #11883: fix boehm-gc so it builds on OS X Lion (John Palmieri)
- #11885: call function in sage.libs.mpmath.utils doesn't handle parent=parent(float) (Benjamin Jones)
- #11891: NumberField(...).pari_polynomial() should return an integral polynomial (Jeroen Demeyer)
- #11896: MPIR 2.x may fail to build on 32-bit operating systems (Leif Leonhardy, Jeroen Demeyer)
- #11914: `sage -n` fails when current directory is $SAGE_ROOT/devel/sage (Leif Leonhardy)
- #11917: Memory leaks with LP Solvers (Nathann Cohen)
- #11923: Graph Theory: Small fix on show3d for graphs (Fidel Barrera-Cruz)
- #11926: "make" should run Sage once (Jeroen Demeyer)
- #11928: Update Graph.clique_number to use MILP (Nathann Cohen)
- #11944: Update Graph.clique_maximum to use MILP (Nathann Cohen)
- #11959: Remove the necessity to set SAGE_PARALLEL_SPKG_BUILD (John Palmieri)
- #11964: MPIR: Use proper ABI name on Linux PPC64 (Leif Leonhardy)
- #11972: avoid race conditions when creating directories (John Palmieri)
- #8654: add "sage -sqlite3" command line option (R. Andrew Ohana, Jeroen Demeyer)
- #9211: graph vertices cut off (Jason Grout)
- #9334: Implement Hilbert symbols over number fields (Aly Deines, Marco Streng, Jeroen Demeyer)
- #10112: random_prime does not handle erroneous input gracefully - it hangs (Mike Hansen, Francis Clarke)
- #10195: Occasional doctest failure in libs/fplll/fplll.pyx
- #10767: Fractional ideals intersection gives wrong answers in some cases. (Jeroen Demeyer)
- #11130: Update PARI to version 2.5.0 (Jeroen Demeyer, John Cremona)
- #11321: Make lcalc compatible with the new PARI (Jeroen Demeyer, Leif Leonhardy)
- #11555: Expand print representation of free module morphisms (Rob Beezer)
- #11604: PARI qfminim bug on 32-bit systems (Sage 4.7 and 4.6.2) (Jeroen Demeyer)
- #11607: read constraints from linear program (John Perry)
- #11761: Upgrade Cython to 0.15.1 (Robert Bradshaw)
- #11762: More import cleanup (Robert Bradshaw)
- #11836: gens_reduced() does not handle "large" ideals (Jeroen Demeyer)
- #11854: Fix conversion QQ -> PARI (Jeroen Demeyer)
- #11886: fix scipy so it builds on OS X Lion (John Palmieri)
- #11890: Sage cannot factor polynomials over number fields with unfactorable discriminant (Jeroen Demeyer)
- #11911: Fix missing uniqueness of parents when unpickling libsingular polynomial rings (Simon King)
- #11946: Change iteration order for finite field multiplicative_generator() (Jeroen Demeyer)
- #11988: Bug in LaTeX representation of Dirichlet characters (Johan Bosman)
- #11991: record time, version in sage-starts (John Palmieri)
- #11993: Fix output of `sage --version` (Leif Leonhardy)
- #6894: Plot list of functions with different colors by default
- #8239: misleading teichmuller behaviour (David Roe)
- #9408: relativize in number fields is broken
- #9847: Handle a preset R_PROFILE variable
- #10620: Upgrade Sphinx to version 1.1.2 (John Palmieri)
- #11279: Convex Hulls, Hull number in graphs (Nathann Cohen)
- #11721: Silent failures in matrix code (Rob Beezer)
- #11781: Conversion from Zmod element to padic element raises "not a power of the same prime" incorrectly, add conversion from residue_field (Robert Harron)
- #11784: Add more sanity checks to FiniteField constructor (Luis Felipe Tabera Alonso)
- #11927: option degree_sequence for graphs (Hiroto Mulia, John Pang)
- #11937: Doctest fails in gap_reps.py when Kash is installed (Daniel Krenn)
- #11949: Gurobi Support ! (Nathann Cohen)
- #11952: Fix conversion Python -> PARI (Jeroen Demeyer)
- #11965: OS X 10.7 Lion: Error building sage.misc.darwin_utilities extension (William Stein)
- #11966: OS X 10.7 Lion: Maxima fails to build (William Stein)
- #11969: Clean up top-level Makefile (Jeroen Demeyer)
- #11971: get_memory_usage() doctest failure (Jeroen Demeyer)
- #11994: Vertex separation and pathwidth in Sage (Nathann Cohen)
- #12000: block_matrix over cyclotomics fails (Volker Braun)
- #12002: avoid race conditions when creating directories - scripts (Jeroen Demeyer)
- #12026: root finding misses a root over QQbar (William Stein)
- #12029: Fast conversion of ClonableIntArray to list (Simon King, Florent Hivert)
- #12031: Remove sage-sage.py (Jeroen Demeyer)
- #12036: improve doctest coverage of gsl/interpolation.pyx from 0% to 100% (William Stein)
- #12037: `search_src` and friends shouldn't look in hidden files (Dan Drake)
- #12038: Complex numbers can segfault if given bad input to the __init__ method (David Roe)
- #1236: tate pairings on elliptic curves -- add to sage
- #4260: use LinBox as native matrix representation for dense matrices over GF(p) (Burcin Erocal, Martin Albrecht, Rob Beezer)
- #5177: Notebook keeps directories for deleted cells (Tim Dumol, Mitesh Patel)
- #5852: Properly canonicalize $SAGE_ROOT (Jeroen Demeyer)
- #6124: Bug in galois_group of a p-adic field extension
- #7929: Pickling fails for some residue fields
- #8274: R's spkg-install needs a good overhaul.
- #9562: Add M4RIE to Sage (Martin Albrecht)
- #9751: Documentation for AdditiveMagmas' addition_table (Rob Beezer)
- #11414: CartesianProduct should return a list of tuples
- #11583: Sagetex example has some nontrivial dependancy's (Maarten Derickx, Dan Drake)
- #11650: Make 'convert' the standard way to produce animated gifs (again) (John Palmieri)
- #11796: inconsistency between 0.nbits() and 0.ndigits(base=2) (Paul Zimmermann)
- #11826: coercing to the residue field of p-adic extensions
- #11847: unexpexted behavior of degree() with matrix ordering (John Perry)
- #11884: Fix ECL so it builds on OS X Lion (Dmitrii Pasechnik, Mike Hansen, Karl-Dieter Crisman, William Stein, John Palmieri, Jeroen Demeyer)
- #11916: add numerical integration to reference manual
- #11924: lazy_import_cache.get_cache_file doesn't work well if DOT_SAGE doesn't end with a slash (John Palmieri)
- #11936: image_type of galois_representation of EllipticCurve hangs on first call (Johan Bosman)
- #11963: aspect ratio is not handled correctly in combined plots (Jason Grout)
- #11997: fix a mistake in the docstring for lift_to_sl2z (etc.) (William Stein)
- #12012: 'make distclean' should clean up start.log, spkg/parallel_make.cfg (John Palmieri)
- #12017: Adds CoerceKey
- #12019: doctest failure in linear_programming.rst (multiple correct solutions) (Nathann Cohen)
- #12020: bug in zero_matrix rewrite in matrix_space.py (Martin Albrecht, Volker Braun, Jeroen Demeyer)
- #12022: failing doctest in integer_vector.py (gale ryser theorem) / multiple solutions (Nathann Cohen)
- #12041: Replace __getslice__ with functionality in __getitem__ in several files (André Apitzsch)
- #12047: numerical_integral(f, a, a) should always be zero (Jeroen Demeyer)
- #12049: matrix_integer_dense.pyx: 'algorithn' instead of 'algorithm' (Daniel Krenn)
- #12052: Some distance computations remained *slow* (Nathann Cohen)
- #12056: sage -sdist/-bdist should use the current branch (Jeroen Demeyer)
- #12058: Mercurial should not enable pager by default (Jeroen Demeyer)
- #12079: Two memory leaks in mpmath extension code (Fredrik Johansson)
- #12081: Do not include doc/output in sage spkg (Jeroen Demeyer)
- #12082: spkg-install of extcode, sage_scripts should preserve timestamp (Jeroen Demeyer)
- #12085: Fix docstring in sage/crypto/boolean_function.pyx (Jeroen Demeyer, André Apitzsch)
- #12086: Do not include doc/en/reference/sage in sage spkg (Jeroen Demeyer)
- #12087: Clean up devel/sage/.hgignore (Jeroen Demeyer)
- #12097: General clean-up of sage-make_relative (Jeroen Demeyer)
- #12098: Readline should depend on termcap (Jeroen Demeyer)
- #879: "sage -testall" should summarize all failures at the end of the run
- #3447: sage -t foo gives wrong path to the file foo in the output
- #5352: the valgrind log files in sage-doctest are written to $HOME/.sage instead of $DOT_SAGE (André Apitzsch)
- #8525: mistake in docstring for R=Zp(3)'s R.plot method. (David Roe)
- #10292: padics, precision, list() and printing (David Roe)
- #10967: R breaks if SAGE_LOCAL undefined
- #11260: ECL-11.1.1 fails to compile maxima on WIndows 7 / Cygwin 1.7.9
- #11526: count_points() error for generic schemes. (includes patch) (Miguel Marco, David Eklund, Frithjof Schulze)
- #11556: Linear transformations, built from free module morphisms (Rob Beezer)
- #11735: Bug in is_chordal (Nathann Cohen)
- #11782: Bug in discriminant of polynomials over Z/nZ with n composite (Johan Bosman)
- #11933: Fix qepcad doctests (Jeroen Demeyer)
- #11987: integrate returns divergent, maxima.integrate the correct result (Nils Bruin, Karl-Dieter Crisman)
- #12035: `list_plot` should accept lists of complex numbers (Dan Drake)
- #12057: Upgrade R (r-project.org) (Volker Braun)
- #12063: EllipticCurve._pari_() should use .pari_curve() (Jeroen Demeyer)
- #12065: Failing doctest - R installation broken?
- #12096: Disable parallel "make install" in Python spkg (Jeroen Demeyer)
- #12104: Cygwin import problem with randstate
- #12106: Implement sage --rsyncdist (Jeroen Demeyer)
- #12111: Bug in Integer.is_power_of (David Roe)
- #12122: Add doctest to check sage_make_relative (Jeroen Demeyer)
- #12126: Fix upgrading of boehm-gc package (John Palmieri)
- #12128: "hg verify" is not a proper way to check whether there is an hg repo (Jeroen Demeyer)
- #12137: Fix parallel building of Singular (Jeroen Demeyer)
- #12138: Fix parallel building zlib (Jeroen Demeyer)
- #12139: Disable parallel "make install" in MPIR spkg (Jeroen Demeyer)
- #2220: irreducibility testing in relative extensions seems to be messed up (Michael Orlitzky)
- #6087: graph automorphism group segfaults on invalid input (fixed already; add doctest) (Michael Orlitzky)
- #6424: One can no longer convert I=sqrt(-1) to sympy (Ondrej Certik, Douglas McNeil)
- #8078: Fix algsys in Maxima (Michael Orlitzky)
- #8168: Keyword option to make reset() detach all attached files (Mitesh Patel)
- #9786: Subsets(list, submultiset=True): wrong output (Lukáš Lánský)
- #10172: Rational Point algorithm bug (Moritz Minzlaff)
- #10621: Make Conic class able to find points over number fields (Marco Streng)
- #10642: Additional points in a documentation titles (Ivo Hedtke, Julian Rueth)
- #10791: Fix and upgrade Gram-Schmidt (Rob Beezer)
- #11597: Dimension of the singular set (Volker Braun)
- #11637: Updated docstring for Graph.cores() (Nathann Cohen)
- #11758: Bug in global_height function (David Krumm, Maarten Derickx)
- #11799: typo in tiny_integrals (André Apitzsch)
- #11899: Add #long time at various places (Jeroen Demeyer)
- #11904: Change default variable name for NumberField and NumberFieldElement -> PARI conversion (Jeroen Demeyer)
- #11947: Implement cardinality for polynomial quotient rings (Jeroen Demeyer)
- #11950: Bug in topological_sort (Nathann Cohen)
- #11961: Fixes a bug in is_chordal -- two algorithms (Nathann Cohen, Jan Elffers)
- #11976: matrix plot colorbars (Jason Grout)
- #12003: improving the list of command-line options in the Sage reference manual (John Palmieri)
- #12016: parallelism in Sage: just use value of 'MAKE' (John Palmieri, Jeroen Demeyer)
- #12108: is_eulerian doesn't handle disconnected graphs properly (Lukáš Lánský)
- #12118: Fix parallel building of libgpg_error (Jeroen Demeyer, Justin Walker)
- #12131: $SAGE_LOCAL/lib and lib64 (Volker Braun, Simon King)
- #12146: hidden markov models: add option to sample/generate to specify the starting state of sequence (William Stein)
- #12157: Segfault in __Pyx_check_binary_version (François Bissey)
- #12158: Segfault in PARI's err_init() during pari_init_opts() (Jeroen Demeyer)
- #12164: graph subgraph_search does not return copy of subgraph (Jason Grout)
- #12166: Elliptic curves: improve documentation of period_lattice (Johan Bosman)
- #12176: Compute Minkowski bound for relative number fields (Michiel Kosters)
- #12213: aspect_ratio and figsize for graphics do not work as expected in sage-4.8.alpha5 (William Stein)
- #8029: Defect: Power series over a polynomial ring with real coefficients (Eduardo Dueñez, Michael Orlitzky)
- #9419: Update Developers Guide to state how patches should be made. (John Palmieri)
- #11245: broken tar on cygwin (Windows 7), cddlib and networkx fail to install (Karl-Dieter Crisman, Dmitrii Pasechnik)
- #11319: Cannot create homomorphism from prime residue field to finite field (Johan Bosman)
- #11417: binomial of polynomial is not polynomial (Sam Scott)
- #11704: Resolve symbolic links in $HOME/.sage (Jeroen Demeyer)
- #11838: multivariate factorization over non-prime finite fields hangs (Paul Zimmermann)
- #11877: complete_primary_decomposition should be available as primary_decomposition_complete (Niels Ranosch)
- #11902: Fan subdivision should check rays (Andrey Novoseltsev)
- #11948: Fix numeric evaluation of error function (Jeroen Demeyer)
- #12066: delete databases/gamma0wt2.py (R. Andrew Ohana)
- #12067: delete databases/bz2Pickle.py (R. Andrew Ohana)
- #12124: Fix Eisenstein L-series precision (Jeroen Demeyer)
- #12147: Binary NTL Finite field elements can't coerce to GF(2) (Johan Sebastian Rosenkilde Nielsen)
- #12156: Pretty print LatexExpr directly (Jeroen Demeyer, Punarbasu Purkayastha)
- #12178: Error in typeset of polynomials (Punarbasu Purkayastha)
- #12187: Bug in is_singular for plane projective curves (Charlie Turner)
- #12190: Remove useless scripts from local/bin (Maarten Derickx)
- #12191: is_squarefree for integer polynomials (Michiel Kosters)
- #12196: Improve latex for quadratic fields (Colton Pauderis)
- #12197: Improve LaTeX of differential form elements (Jeroen Demeyer)
- #12207: Typing "sage -cython" doesn't work due to bug in wrapper script (William Stein, Jeroen Demeyer)
- #12208: LU decomposition gives wrong results on cyclotomic matrices (MacOSX-specific) (Rob Beezer)
- #12211: bug in equation checking for quasi projective/affine schemes (David Eklund)
- #1634: update zodb to 3.8.1
- #5323: "./sage -t" gives misleading error message when doctesting non-existing file with absolute patch
- #5938: graph plotting -- ploting of graphs (networks) is somehow messed up/scaled wrong/cropped wrong since it doesn't work with graphics_array
- #5970: Weak references in Polynomial Ring cache
- #6439: doctests beginning with Sage: are silently ignored
- #8245: tutorial: typo in section "Euler’s Method for Systems of Differential Equations" (Kenneth Smith)
- #10965: Fix trivial problems in plot/polygon (Peter Story, Kenneth Smith)
- #11325: Multiple line cells generation in live help broken, lots of samples unusable
- #11497: Twisted won't build on Cygwin - sometimes
- #11502: Maxima fails to build on Cygwin 1.7.3 on XP
- #11724: only print warnings when needed
- #11970: r-2.10.1.p4 may fail to configure with readline (BLOCKS Sage from building on Ubuntu 11.10) (Leif Leonhardy)
- #12072: Wrap pynac ex.normal method
- #12110: [ARM] Singular 3-1-3-3.p1 doesn't compile as-is (Julien Puydt)
- #12130: CPLEX acquires maximal number of threads on system
- #12149: float('nan')>1 crashes Sage. (William Stein)
- #12163: Missing digits in numerical_approx
- #12185: Bug in norm for orders of relative number fields
- #12206: Put SAGE_SCRIPTS first in spkg/standard/deps (William Stein)
- #12222: default aspect ratio for geometric objects (Jason Grout)
- #12245: M4RIE test suite fails due to missing dependencies (Martin Albrecht, David Loeffler)
- #12248: Do not run "make -q" when upgrading (Jeroen Demeyer)
- #12249: Add doctest for sage-run (Jeroen Demeyer)
- #12252: latex(number field) sometimes looks stupid (though is technically correct)
- #12258: Prune .hgtags files (Keshav Kini)
- #12263: Unset TERM when running sage non-interactively (Jeroen Demeyer)
- #12274: Typo in FreeModuleElement.norm (Eviatar Bach)
- #12282: Fix strcmp() with NULL argument in termcap library (Jeroen Demeyer)
- #12283: Numerical noise in chmm.pyx (Jeroen Demeyer)
- #12285: Update darwin_memory_usage.c for Lion header files
- #12288: hg_sage and friends should unset HGPLAIN (John Palmieri)
- #12312: Disable parallel building of ATLAS (Jeroen Demeyer)
- Mac OS X 10.6 or later
- 64-bit processor
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