The Interactive Shell#

In most of this tutorial, we assume you start the Sage interpreter using the sage command. This starts a customized version of the IPython shell, and imports many functions and classes, so they are ready to use from the command prompt. Further customization is possible by editing the $SAGE_ROOT/ipythonrc file. Upon starting Sage, you get output similar to the following:

│ SageMath version 9.7, Release Date: 2022-01-10                     │
│ Using Python 3.10.4. Type "help()" for help.                       │


To quit Sage either press Ctrl-D or type quit or exit.

sage: quit
Exiting Sage (CPU time 0m0.00s, Wall time 0m0.89s)

The wall time is the time that elapsed on the clock hanging from your wall. This is relevant, since CPU time does not track time used by subprocesses like GAP or Singular.

(Avoid killing a Sage process with kill -9 from a terminal, since Sage might not kill child processes, e.g., Maple processes, or cleanup temporary files from $HOME/.sage/tmp.)

Your Sage Session#

The session is the sequence of input and output from when you start Sage until you quit. Sage logs all Sage input, via IPython. In fact, if you’re using the interactive shell (not the notebook interface), then at any point you may type %history (or %hist) to get a listing of all input lines typed so far. You can type ? at the Sage prompt to find out more about IPython, e.g., “IPython offers numbered prompts … with input and output caching. All input is saved and can be retrieved as variables (besides the usual arrow key recall). The following GLOBAL variables always exist (so don’t overwrite them!)”:

_:  previous input (interactive shell and notebook)
__: next previous input (interactive shell only)
_oh : list of all inputs (interactive shell only)

Here is an example:

sage: factor(100)
 _1 = 2^2 * 5^2
sage: kronecker_symbol(3,5)
 _2 = -1
sage: %hist   #This only works from the interactive shell, not the notebook.
1: factor(100)
2: kronecker_symbol(3,5)
3: %hist
sage: _oh
 _4 = {1: 2^2 * 5^2, 2: -1}
sage: _i1
 _5 = 'factor(ZZ(100))\n'
sage: eval(_i1)
 _6 = 2^2 * 5^2
sage: %hist
1: factor(100)
2: kronecker_symbol(3,5)
3: %hist
4: _oh
5: _i1
6: eval(_i1)
7: %hist

We omit the output numbering in the rest of this tutorial and the other Sage documentation.

You can also store a list of input from session in a macro for that session.

sage: E = EllipticCurve([1,2,3,4,5])
sage: M = ModularSymbols(37)
sage: %hist
1: E = EllipticCurve([1,2,3,4,5])
2: M = ModularSymbols(37)
3: %hist
sage: %macro em 1-2
Macro `em` created. To execute, type its name (without quotes).
sage: E
Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over
Rational Field
sage: E = 5
sage: M = None
sage: em
Executing Macro...
sage: E
Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over
Rational Field

When using the interactive shell, any UNIX shell command can be executed from Sage by prefacing it by an exclamation point !. For example,

sage: !ls
auto  example.sage glossary.tex  t  tmp  tut.log  tut.tex

returns the listing of the current directory.

The PATH has the Sage bin directory at the front, so if you run gp, gap, singular, maxima, etc., you get the versions included with Sage.

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 version
sage: !singular
                     SINGULAR                             /  Development
 A Computer Algebra System for Polynomial Computations   /   version 3-0-1
     by: G.-M. Greuel, G. Pfister, H. Schoenemann        \   October 2005
FB Mathematik der Universitaet, D-67653 Kaiserslautern    \

Logging Input and Output#

Logging your Sage session is not the same as saving it (see Saving and Loading Complete Sessions for that). To log input (and optionally output) use the logstart command. Type logstart? for more details. You can use this command to log all input you type, all output, and even play back that input in a future session (by simply reloading the log file).

was@form:~$ sage
│ SageMath version 9.7, Release Date: 2022-01-10                     │
│ Using Python 3.10.4. Type "help()" for help.                       │

sage: logstart setup
Activating auto-logging. Current session state plus future input saved.
Filename       : setup
Mode           : backup
Output logging : False
Timestamping   : False
State          : active
sage: E = EllipticCurve([1,2,3,4,5]).minimal_model()
sage: F = QQ^3
sage: x,y = QQ['x,y'].gens()
sage: G = E.gens()
Exiting Sage (CPU time 0m0.61s, Wall time 0m50.39s).
was@form:~$ sage
│ SageMath version 9.7, Release Date: 2022-01-10                     │
│ Using Python 3.10.4. Type "help()" for help.                       │

sage: load("setup")
Loading log file <setup> one line at a time...
Finished replaying log file <setup>
sage: E
Elliptic Curve defined by y^2 + x*y  = x^3 - x^2 + 4*x + 3 over Rational
sage: x*y
sage: G
[(2 : 3 : 1)]

If you use Sage in the Linux KDE terminal konsole then you can save your session as follows: after starting Sage in konsole, select “settings”, then “history…”, then “set unlimited”. When you are ready to save your session, select “edit” then “save history as…” and type in a name to save the text of your session to your computer. After saving this file, you could then load it into an editor, such as xemacs, and print it.

Paste Ignores Prompts#

Suppose you are reading a session of Sage or Python computations and want to copy them into Sage. But there are annoying >>> or sage: prompts to worry about. In fact, you can copy and paste an example, including the prompts if you want, into Sage. In other words, by default the Sage parser strips any leading >>> or sage: prompt before passing it to Python. For example,

sage: 2^10
sage: sage: sage: 2^10
sage: >>> 2^10

Timing Commands#

If you place the %time command at the beginning of an input line, the time the command takes to run will be displayed after the output. For example, we can compare the running time for a certain exponentiation operation in several ways. The timings below will probably be much different on your computer, or even between different versions of Sage. First, native Python:

sage: %time a = int(1938)^int(99484)
CPU times: user 0.66 s, sys: 0.00 s, total: 0.66 s
Wall time: 0.66

This means that 0.66 seconds total were taken, and the “Wall time”, i.e., the amount of time that elapsed on your wall clock, is also 0.66 seconds. If your computer is heavily loaded with other programs, the wall time may be much larger than the CPU time.

It’s also possible to use the timeit function to try to get timing over a large number of iterations of a command. This gives slightly different information, and requires the input of a string with the command you want to time.

sage: timeit("int(1938)^int(99484)")
5 loops, best of 3: 44.8 ms per loop

Next we time exponentiation using the native Sage Integer type, which is implemented (in Cython) using the GMP library:

sage: %time a = 1938^99484
CPU times: user 0.04 s, sys: 0.00 s, total: 0.04 s
Wall time: 0.04

Using the PARI C-library interface:

sage: %time a = pari(1938)^pari(99484)
CPU times: user 0.05 s, sys: 0.00 s, total: 0.05 s
Wall time: 0.05

GMP is better, but only slightly (as expected, since the version of PARI built for Sage uses GMP for integer arithmetic).

You can also time a block of commands using the cputime command, as illustrated below:

sage: t = cputime()
sage: a = int(1938)^int(99484)
sage: b = 1938^99484
sage: c = pari(1938)^pari(99484)
sage: cputime(t)                       # somewhat random output
sage: cputime?
    Return the time in CPU second since Sage started, or with optional
    argument t, return the time since time t.
        t -- (optional) float, time in CPU seconds
        float -- time in CPU seconds

The walltime command behaves just like the cputime command, except that it measures wall time.

We can also compute the above power in some of the computer algebra systems that Sage includes. In each case we execute a trivial command in the system, in order to start up the server for that program. The most relevant time is the wall time. However, if there is a significant difference between the wall time and the CPU time then this may indicate a performance issue worth looking into.

sage: time 1938^99484;
CPU times: user 0.01 s, sys: 0.00 s, total: 0.01 s
Wall time: 0.01
sage: gp(0)
sage: time g = gp('1938^99484')
CPU times: user 0.00 s, sys: 0.00 s, total: 0.00 s
Wall time: 0.04
sage: maxima(0)
sage: time g = maxima('1938^99484')
CPU times: user 0.00 s, sys: 0.00 s, total: 0.00 s
Wall time: 0.30
sage: kash(0)
sage: time g = kash('1938^99484')
CPU times: user 0.00 s, sys: 0.00 s, total: 0.00 s
Wall time: 0.04
sage: mathematica(0)
sage: time g = mathematica('1938^99484')
CPU times: user 0.00 s, sys: 0.00 s, total: 0.00 s
Wall time: 0.03
sage: maple(0)
sage: time g = maple('1938^99484')
CPU times: user 0.00 s, sys: 0.00 s, total: 0.00 s
Wall time: 0.11
sage: gap(0)
sage: time g = gap.eval('1938^99484;;')
CPU times: user 0.00 s, sys: 0.00 s, total: 0.00 s
Wall time: 1.02

Note that GAP and Maxima are the slowest in this test (this was run on the machine Because of the pexpect interface overhead, it is perhaps unfair to compare these to Sage, which is the fastest.

Other IPython tricks#

As noted above, Sage uses IPython as its front end, and so you can use any of IPython’s commands and features. You can read the full IPython documentation. Meanwhile, here are some fun tricks – these are called “Magic commands” in IPython:

  • You can use %edit (or %ed or ed) to open an editor, if you want to type in some complex code. Before you start Sage, make sure that the EDITOR environment variable is set to your favorite editor (by putting export EDITOR=/usr/bin/emacs or export EDITOR=/usr/bin/vim or something similar in the appropriate place, like a .profile file). From the Sage prompt, executing %edit will open up the named editor. Then within the editor you can define a function:

    def some_function(n):
        return n**2 + 3*n + 2

    Save and quit from the editor. For the rest of your Sage session, you can then use some_function. If you want to modify it, type %edit some_function from the Sage prompt.

  • If you have a computation and you want to modify its output for another use, perform the computation and type %rep: this will place the output from the previous command at the Sage prompt, ready for you to edit it.

    sage: f(x) = cos(x)
    sage: f(x).derivative(x)

    At this point, if you type %rep at the Sage prompt, you will get a new Sage prompt, followed by -sin(x), with the cursor at the end of the line.

For more, type %quickref to get a quick reference guide to IPython. As of this writing (April 2011), Sage uses version 0.9.1 of IPython, and the documentation for its magic commands is available online. Various slightly advanced aspects of magic command system are documented here in IPython.

Errors and Exceptions#

When something goes wrong, you will usually see a Python “exception”. Python even tries to suggest what raised the exception. Often you see the name of the exception, e.g., NameError or ValueError (see the Python Library Reference [PyLR] for a complete list of exceptions). For example,

sage: 3_2
   File "<console>", line 1
SyntaxError: invalid ...

sage: EllipticCurve([0,infinity])
Traceback (most recent call last):
TypeError: Unable to coerce Infinity (<class 'sage...Infinity'>) to Rational

The interactive debugger is sometimes useful for understanding what went wrong. You can toggle it on or off using %pdb (the default is off). The prompt ipdb> appears if an exception is raised and the debugger is on. From within the debugger, you can print the state of any local variable, and move up and down the execution stack. For example,

sage: %pdb
Automatic pdb calling has been turned ON
sage: EllipticCurve([1,infinity])
<class 'exceptions.TypeError'>             Traceback (most recent call last)


For a list of commands in the debugger, type ? at the ipdb> prompt:

ipdb> ?

Documented commands (type help <topic>):
EOF    break  commands   debug    h       l     pdef   quit    tbreak
a      bt     condition  disable  help    list  pdoc   r       u
alias  c      cont       down     ignore  n     pinfo  return  unalias
args   cl     continue   enable   j       next  pp     s       up
b      clear  d          exit     jump    p     q      step    w
whatis where

Miscellaneous help topics:
exec  pdb

Undocumented commands:
retval  rv

Type Ctrl-D or quit to return to Sage.

Reverse Search and Tab Completion#

Reverse search: Type the beginning of a command, then Ctrl-p (or just hit the up arrow key) to go back to each line you have entered that begins in that way. This works even if you completely exit Sage and restart later. You can also do a reverse search through the history using Ctrl-r. All these features use the readline package, which is available on most flavors of Linux.

To illustrate tab completion, first create the three dimensional vector space \(V=\QQ^3\) as follows:

sage: V = VectorSpace(QQ,3)
sage: V
Vector space of dimension 3 over Rational Field

You can also use the following more concise notation:

sage: V = QQ^3

Then it is easy to list all member functions for \(V\) using tab completion. Just type V., then type the Tab key on your keyboard:

sage: V.[tab key]

If you type the first few letters of a function, then the Tab key, you get only functions that begin as indicated.

sage: V.i[tab key]
V.is_ambient  V.is_dense    V.is_full     V.is_sparse

If you wonder what a particular function does, e.g., the coordinates function, type V.coordinates? for help or V.coordinates?? for the source code, as explained in the next section.

Integrated Help System#

Sage features an integrated help facility. Type a function name followed by ? for the documentation for that function.

sage: V = QQ^3
sage: V.coordinates?
Type:           instancemethod
Base Class:     <class 'instancemethod'>
String Form:    <bound method FreeModule_ambient_field.coordinates of Vector
space of dimension 3 over Rational Field>
Namespace:      Interactive
File:           /home/was/s/local/lib/python2.4/site-packages/sage/modules/f
Definition:     V.coordinates(self, v)
    Write v in terms of the basis for self.

    Returns a list c such that if B is the basis for self, then

            sum c_i B_i = v.

    If v is not in self, raises an ArithmeticError exception.

        sage: M = FreeModule(IntegerRing(), 2); M0,M1=M.gens()
        sage: W = M.submodule([M0 + M1, M0 - 2*M1])
        sage: W.coordinates(2*M0-M1)
        [2, -1]

As shown above, the output tells you the type of the object, the file in which it is defined, and a useful description of the function with examples that you can paste into your current session. Almost all of these examples are regularly automatically tested to make sure they work and behave exactly as claimed.

Another feature that is very much in the spirit of the open source nature of Sage is that if f is a Python function, then typing f?? displays the source code that defines f. For example,

sage: V = QQ^3
sage: V.coordinates??
Type:           instancemethod
def coordinates(self, v):
        Write $v$ in terms of the basis for self.
        return self.coordinate_vector(v).list()

This tells us that all the coordinates function does is call the coordinate_vector function and change the result into a list. What does the coordinate_vector function do?

sage: V = QQ^3
sage: V.coordinate_vector??
def coordinate_vector(self, v):
        return self.ambient_vector_space()(v)

The coordinate_vector function coerces its input into the ambient space, which has the effect of computing the vector of coefficients of \(v\) in terms of \(V\). The space \(V\) is already ambient since it’s just \(\QQ^3\). There is also a coordinate_vector function for subspaces, and it’s different. We create a subspace and see:

sage: V = QQ^3; W = V.span_of_basis([V.0, V.1])
sage: W.coordinate_vector??
def coordinate_vector(self, v):
        # First find the coordinates of v wrt echelon basis.
        w = self.echelon_coordinate_vector(v)
        # Next use transformation matrix from echelon basis to
        # user basis.
        T = self.echelon_to_user_matrix()
        return T.linear_combination_of_rows(w)

(If you think the implementation is inefficient, please sign up to help optimize linear algebra.)

You may also type help(command_name) or help(class) for a manpage-like help file about a given class.

sage: help(VectorSpace)
Help on class VectorSpace ...

class VectorSpace(__builtin__.object)
 |  Create a Vector Space.
 |  To create an ambient space over a field with given dimension
 |  using the calling syntax ...

When you type q to exit the help system, your session appears just as it was. The help listing does not clutter up your session, unlike the output of function_name? sometimes does. It’s particularly helpful to type help(module_name). For example, vector spaces are defined in sage.modules.free_module, so type help(sage.modules.free_module) for documentation about that whole module. When viewing documentation using help, you can search by typing / and in reverse by typing ?.

Saving and Loading Individual Objects#

Suppose you compute a matrix or worse, a complicated space of modular symbols, and would like to save it for later use. What can you do? There are several approaches that computer algebra systems take to saving individual objects.

  1. Save your Game: Only support saving and loading of complete sessions (e.g., GAP, Magma).

  2. Unified Input/Output: Make every object print in a way that can be read back in (GP/PARI).

  3. Eval: Make it easy to evaluate arbitrary code in the interpreter (e.g., Singular, PARI).

Because Sage uses Python, it takes a different approach, which is that every object can be serialized, i.e., turned into a string from which that object can be recovered. This is in spirit similar to the unified I/O approach of PARI, except it doesn’t have the drawback that objects print to screen in too complicated of a way. Also, support for saving and loading is (in most cases) completely automatic, requiring no extra programming; it’s simply a feature of Python that was designed into the language from the ground up.

Almost all Sage objects x can be saved in compressed form to disk using save(x, filename) (or in many cases To load the object back in, use load(filename).

sage: A = MatrixSpace(QQ,3)(range(9))^2
sage: A
[ 15  18  21]
[ 42  54  66]
[ 69  90 111]
sage: save(A, 'A')

You should now quit Sage and restart. Then you can get A back:

sage: A = load('A')
sage: A
[ 15  18  21]
[ 42  54  66]
[ 69  90 111]

You can do the same with more complicated objects, e.g., elliptic curves. All data about the object that is cached is stored with the object. For example,

sage: E = EllipticCurve('11a')
sage: v = E.anlist(100000)              # takes a while
sage: save(E, 'E')
sage: quit

The saved version of E takes 153 kilobytes, since it stores the first 100000 \(a_n\) with it.

~/tmp$ ls -l E.sobj
-rw-r--r--  1 was was 153500 2006-01-28 19:23 E.sobj
~/tmp$ sage [...]
sage: E = load('E')
sage: v = E.anlist(100000)              # instant!

(In Python, saving and loading is accomplished using the cPickle module. In particular, a Sage object x can be saved via cPickle.dumps(x, 2). Note the 2!)

Sage cannot save and load individual objects created in some other computer algebra systems, e.g., GAP, Singular, Maxima, etc. They reload in a state marked “invalid”. In GAP, though many objects print in a form from which they can be reconstructed, many don’t, so reconstructing from their print representation is purposely not allowed.

sage: a = gap(2)
sage: load('a')
Traceback (most recent call last):
ValueError: The session in which this object was defined is no longer

GP/PARI objects can be saved and loaded since their print representation is enough to reconstruct them.

sage: a = gp(2)
sage: load('a')

Saved objects can be re-loaded later on computers with different architectures or operating systems, e.g., you could save a huge matrix on 32-bit OS X and reload it on 64-bit Linux, find the echelon form, then move it back. Also, in many cases you can even load objects into versions of Sage that are different than the versions they were saved in, as long as the code for that object isn’t too different. All the attributes of the objects are saved, along with the class (but not source code) that defines the object. If that class no longer exists in a new version of Sage, then the object can’t be reloaded in that newer version. But you could load it in an old version, get the objects dictionary (with x.__dict__), and save the dictionary, and load that into the newer version.

Saving as Text#

You can also save the ASCII text representation of objects to a plain text file by simply opening a file in write mode and writing the string representation of the object (you can write many objects this way as well). When you’re done writing objects, close the file.

sage: R.<x,y> = PolynomialRing(QQ,2)
sage: f = (x+y)^7
sage: o = open('file.txt','w')
sage: o.write(str(f))
sage: o.close()

Saving and Loading Complete Sessions#

Sage has very flexible support for saving and loading complete sessions.

The command save_session(sessionname) saves all the variables you’ve defined in the current session as a dictionary in the given sessionname. (In the rare case when a variable does not support saving, it is simply not saved to the dictionary.) The resulting file is an .sobj file and can be loaded just like any other object that was saved. When you load the objects saved in a session, you get a dictionary whose keys are the variables names and whose values are the objects.

You can use the load_session(sessionname) command to load the variables defined in sessionname into the current session. Note that this does not wipe out variables you’ve already defined in your current session; instead, the two sessions are merged.

First we start Sage and define some variables.

sage: E = EllipticCurve('11a')
sage: M = ModularSymbols(37)
sage: a = 389
sage: t = M.T(2003).matrix(); t.charpoly().factor()
 _4 = (x - 2004) * (x - 12)^2 * (x + 54)^2

Next we save our session, which saves each of the above variables into a file. Then we view the file, which is about 3K in size.

sage: save_session('misc')
Saving a
Saving M
Saving t
Saving E
sage: quit
was@form:~/tmp$ ls -l misc.sobj
-rw-r--r--  1 was was 2979 2006-01-28 19:47 misc.sobj

Finally we restart Sage, define an extra variable, and load our saved session.

sage: b = 19
sage: load_session('misc')
Loading a
Loading M
Loading E
Loading t

Each saved variable is again available. Moreover, the variable b was not overwritten.

sage: M
Full Modular Symbols space for Gamma_0(37) of weight 2 with sign 0
and dimension 5 over Rational Field
sage: E
Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational
sage: b
sage: a