Coding in Python for Sage

This chapter discusses some issues with, and advice for, coding in Sage.


If you are planning to develop some new code for Sage, design is important. So think about what your program will do and how that fits into the structure of Sage. In particular, much of Sage is implemented in the object-oriented language Python, and there is a hierarchy of classes that organize code and functionality. For example, if you implement elements of a ring, your class should derive from sage.structure.element.RingElement, rather than starting from scratch. Try to figure out how your code should fit in with other Sage code, and design it accordingly.

Special Sage Functions

Functions with leading and trailing double underscores __XXX__ are all predefined by Python. Functions with leading and trailing single underscores _XXX_ are defined for Sage. Functions with a single leading underscore are meant to be semi-private, and those with a double leading underscore are considered really private. Users can create functions with leading and trailing underscores.

Just as Python has many standard special methods for objects, Sage also has special methods. They are typically of the form _XXX_. In a few cases, the trailing underscore is not included, but this will eventually be changed so that the trailing underscore is always included. This section describes these special methods.

All objects in Sage should derive from the Cython extension class SageObject:

from sage.ext.sage_object import SageObject

class MyClass(SageObject,...):

or from some other already existing Sage class:

from sage.rings.ring import Algebra

class MyFavoriteAlgebra(Algebra):

You should implement the _latex_ and _repr_ method for every object. The other methods depend on the nature of the object.

LaTeX Representation

Every object x in Sage should support the command latex(x), so that any Sage object can be easily and accurately displayed via LaTeX. Here is how to make a class (and therefore its instances) support the command latex.

  1. Define a method _latex_(self) that returns a LaTeX representation of your object. It should be something that can be typeset correctly within math mode. Do not include opening and closing $’s.
  2. Often objects are built up out of other Sage objects, and these components should be typeset using the latex function. For example, if c is a coefficient of your object, and you want to typeset c using LaTeX, use latex(c) instead of c._latex_(), since c might not have a _latex_ method, and latex(c) knows how to deal with this.
  3. Do not forget to include a docstring and an example that illustrates LaTeX generation for your object.
  4. You can use any macros included in amsmath, amssymb, or amsfonts, or the ones defined in SAGE_ROOT/doc/commontex/macros.tex.

An example template for a _latex_ method follows:

class X:
   def _latex_(self):
       Return the LaTeX representation of X.


           sage: a = X(1,2)
           sage: latex(a)
       return '\\frac{%s}{%s}'%(latex(self.numer), latex(self.denom))

As shown in the example, latex(a) will produce LaTeX code representing the object a. Calling view(a) will display the typeset version of this.

Matrix or Vector from Object

Provide a _matrix_ method for an object that can be coerced to a matrix over a ring \(R\). Then the Sage function matrix will work for this object.

The following is from SAGE_ROOT/src/sage/graphs/

class GenericGraph(SageObject):
    def _matrix_(self, R=None):
        if R is None:

    def adjacency_matrix(self, sparse=None, boundary_first=False):

Similarly, provide a _vector_ method for an object that can be coerced to a vector over a ring \(R\). Then the Sage function vector will work for this object. The following is from the file SAGE_ROOT/sage/sage/modules/free_module_element.pyx:

cdef class FreeModuleElement(element_Vector):   # abstract base class
    def _vector_(self, R):
        return self.change_ring(R)

Sage Preparsing

To make Python even more usable interactively, there are a number of tweaks to the syntax made when you use Sage from the commandline or via the notebook (but not for Python code in the Sage library). Technically, this is implemented by a preparse() function that rewrites the input string. Most notably, the following replacements are made:

  • Sage supports a special syntax for generating rings or, more generally, parents with named generators:

    sage: R.<x,y> = QQ[]
    sage: preparse('R.<x,y> = QQ[]')
    "R = QQ['x, y']; (x, y,) = R._first_ngens(2)"
  • Integer and real literals are Sage integers and Sage floating point numbers. For example, in pure Python these would be an attribute error:

    sage: 16.sqrt()
    sage: 87.factor()
    3 * 29
  • Raw literals are not preparsed, which can be useful from an efficiency point of view. Just like Python ints are denoted by an L, in Sage raw integer and floating literals are followed by an “r” (or “R”) for raw, meaning not preparsed. For example:

    sage: a = 393939r
    sage: a
    sage: type(a)
    <... 'int'>
    sage: b = 393939
    sage: type(b)
    <type 'sage.rings.integer.Integer'>
    sage: a == b
  • Raw literals can be very useful in certain cases. For instance, Python integers can be more efficient than Sage integers when they are very small. Large Sage integers are much more efficient than Python integers since they are implemented using the GMP C library.

Consult the file for more details about Sage preparsing, more examples involving raw literals, etc.

When a file foo.sage is loaded or attached in a Sage session, a preparsed version of foo.sage is created with the name The beginning of the preparsed file states:

This file was *autogenerated* from the file foo.sage.

You can explicitly preparse a file with the --preparse command-line option: running

sage --preparse foo.sage

creates the file

The following files are relevant to preparsing in Sage:

  1. SAGE_ROOT/src/bin/sage
  2. SAGE_ROOT/src/bin/sage-preparse
  3. SAGE_ROOT/src/sage/repl/

In particular, the file contains the Sage preparser code.

The Sage Coercion Model

The primary goal of coercion is to be able to transparently do arithmetic, comparisons, etc. between elements of distinct sets. For example, when one writes \(3 + 1/2\), one wants to perform arithmetic on the operands as rational numbers, despite the left term being an integer. This makes sense given the obvious and natural inclusion of the integers into the rational numbers. The goal of the coercion system is to facilitate this (and more complicated arithmetic) without having to explicitly map everything over into the same domain, and at the same time being strict enough to not resolve ambiguity or accept nonsense.

The coercion model for Sage is described in detail, with examples, in the Coercion section of the Sage Reference Manual.


Parent structures (e.g. rings, fields, matrix spaces, etc.) should be immutable and globally unique whenever possible. Immutability means, among other things, that properties like generator labels and default coercion precision cannot be changed.

Global uniqueness while not wasting memory is best implemented using the standard Python weakref module, a factory function, and module scope variable.

Certain objects, e.g. matrices, may start out mutable and become immutable later. See the file SAGE_ROOT/src/sage/structure/

The __hash__ Special Method

Here is the definition of __hash__ from the Python reference manual:

Called by built-in function hash() and for operations on members of hashed collections including set, frozenset, and dict. __hash__() should return an integer. The only required property is that objects which compare equal have the same hash value; it is advised to somehow mix together (e.g. using exclusive or) the hash values for the components of the object that also play a part in comparison of objects. If a class does not define a __cmp__() method it should not define a __hash__() operation either; if it defines __cmp__() or __eq__() but not __hash__(), its instances will not be usable as dictionary keys. If a class defines mutable objects and implements a __cmp__() or __eq__() method, it should not implement __hash__(), since the dictionary implementation requires that a key’s hash value is immutable (if the object’s hash value changes, it will be in the wrong hash bucket).

Notice the phrase, “The only required property is that objects which compare equal have the same hash value.” This is an assumption made by the Python language, which in Sage we simply cannot make (!), and violating it has consequences. Fortunately, the consequences are pretty clearly defined and reasonably easy to understand, so if you know about them they do not cause you trouble. The following example illustrates them pretty well:

sage: v = [Mod(2,7)]
sage: 9 in v
sage: v = set([Mod(2,7)])
sage: 9 in v
sage: 2 in v
sage: w = {Mod(2,7):'a'}
sage: w[2]
sage: w[9]
Traceback (most recent call last):
KeyError: 9

Here is another example:

sage: R = RealField(10000)
sage: a = R(1) + R(10)^-100
sage: a == RDF(1)  # because the a gets coerced down to RDF

but hash(a) should not equal hash(1).

Unfortunately, in Sage we simply cannot require

(#)   "a == b ==> hash(a) == hash(b)"

because serious mathematics is simply too complicated for this rule. For example, the equalities z == Mod(z, 2) and z == Mod(z, 3) would force hash() to be constant on the integers.

The only way we could “fix” this problem for good would be to abandon using the == operator for “Sage equality”, and implement Sage equality as a new method attached to each object. Then we could follow Python rules for == and our rules for everything else, and all Sage code would become completely unreadable (and for that matter unwritable). So we just have to live with it.

So what is done in Sage is to attempt to satisfy (#) when it is reasonably easy to do so, but use judgment and not go overboard. For example,

sage: hash(Mod(2,7))

The output 2 is better than some random hash that also involves the moduli, but it is of course not right from the Python point of view, since 9 == Mod(2,7). The goal is to make a hash function that is fast, but within reason respects any obvious natural inclusions and coercions.


Please avoid catch-all code like this:

except:               # bad

If you do not have any exceptions explicitly listed (as a tuple), your code will catch absolutely anything, including ctrl-C, typos in the code, and alarms, and this will lead to confusion. Also, this might catch real errors which should be propagated to the user.

To summarize, only catch specific exceptions as in the following example:

    return self.__coordinate_ring
except (AttributeError, OtherExceptions) as msg:           # good

Note that the syntax in except is to list all the exceptions that are caught as a tuple, followed by an error message.


We mention two issues with importing: circular imports and importing large third-party modules.

First, you must avoid circular imports. For example, suppose that the file SAGE_ROOT/src/sage/algebras/ started with a line:

from sage.sage.algebras.steenrod_algebra_bases import *

and that the file SAGE_ROOT/src/sage/algebras/ started with a line:

from sage.sage.algebras.steenrod_algebra import SteenrodAlgebra

This sets up a loop: loading one of these files requires the other, which then requires the first, etc.

With this set-up, running Sage will produce an error:

Exception exceptions.ImportError: 'cannot import name SteenrodAlgebra'
in 'sage.rings.polynomial.polynomial_element.
Polynomial_generic_dense.__normalize' ignored
ImportError                       Traceback (most recent call last)

ImportError: cannot import name SteenrodAlgebra

Instead, you might replace the import * line at the top of the file by more specific imports where they are needed in the code. For example, the basis method for the class SteenrodAlgebra might look like this (omitting the documentation string):

def basis(self, n):
    from steenrod_algebra_bases import steenrod_algebra_basis
    return steenrod_algebra_basis(n, basis=self._basis_name,

Second, do not import at the top level of your module a third-party module that will take a long time to initialize (e.g. matplotlib). As above, you might instead import specific components of the module when they are needed, rather than at the top level of your file.

It is important to try to make from sage.all import * as fast as possible, since this is what dominates the Sage startup time, and controlling the top-level imports helps to do this. One important mechanism in Sage are lazy imports, which don’t actually perform the import but delay it until the object is actually used. See sage.misc.lazy_import for more details of lazy imports, and Files and Directory Structure for an example using lazy imports for a new module.


When making a backward-incompatible modification in Sage, the old code should keep working and display a message indicating how it should be updated/written in the future. We call this a deprecation.


Deprecated code can only be removed one year after the first stable release in which it appeared.

Each deprecation warning contains the number of the trac ticket that defines it. We use 666 in the examples below. For each entry, consult the function’s documentation for more information on its behaviour and optional arguments.

  • Rename a keyword: by decorating a function/method with rename_keyword, any user calling my_function(my_old_keyword=5) will see a warning:

    from sage.misc.decorators import rename_keyword
    @rename_keyword(deprecation=666, my_old_keyword='my_new_keyword')
    def my_function(my_new_keyword=True):
        return my_new_keyword
  • Rename a function/method: call deprecated_function_alias() to obtain a copy of a function that raises a deprecation warning:

    from sage.misc.superseded import deprecated_function_alias
    def my_new_function():
    my_old_function = deprecated_function_alias(666, my_new_function)
  • Moving an object to a different module: if you rename a source file or move some function (or class) to a different file, it should still be possible to import that function from the old module. This can be done using a lazy_import() with deprecation. In the old module, you would write:

    from sage.misc.lazy_import import lazy_import
    lazy_import('', 'name_of_the_function', deprecation=666)

    You can also lazily import everything using * or a few functions using a tuple:

    from sage.misc.lazy_import import lazy_import
    lazy_import('', '*', deprecation=666)
    lazy_import('sage.other.module', ('func1', 'func2'), deprecation=666)
  • Remove a name from the global namespace: the function deprecated_callable_import() imports an object into the global namespace. Any user who calls it sees a message inviting him to import the object manually:

    from sage.misc.superseded import deprecated_callable_import

    Alternatively, a lazy_import with deprecation would also work in this case.

  • Any other case: if none of the cases above apply, call deprecation() in the function that you want to deprecate. It will display the message of your choice (and interact properly with the doctest framework):

    from sage.misc.superseded import deprecation
    deprecation(666, "Do not use your computer to compute 1+1. Use your brain.")

Experimental/Unstable Code

You can mark your newly created code (classes/functions/methods) as experimental/unstable. In this case, no deprecation warning is needed when changing this code, its functionality or its interface.

This should allow you to put your stuff in Sage early, without worrying about making (design) changes later.

When satisfied with the code (when stable for some time, say, one year), you can delete this warning.

As usual, all code has to be fully doctested and go through our reviewing process.

  • Experimental function/method: use the decorator experimental. Here is an example:

    from sage.misc.superseded import experimental
    def experimental_function():
        # do something
  • Experimental class: use the decorator experimental for its __init__. Here is an example:

    from sage.misc.superseded import experimental
    class experimental_class(SageObject):
        def __init__(self, some, arguments):
            # do something
  • Any other case: if none of the cases above apply, call experimental_warning() in the code where you want to warn. It will display the message of your choice:

    from sage.misc.superseded import experimental_warning
    experimental_warning(66666, 'This code is not foolproof.')

Using Optional Packages

If a function requires an optional package, that function should fail gracefully—perhaps using a try-except block—when the optional package is not available, and should give a hint about how to install it. For example, typing sage -optional gives a list of all optional packages, so it might suggest to the user that they type that. The command optional_packages() from within Sage also returns this list.