Coding in Python for Sage#

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

Python language standard#

Sage library code needs to be compatible with all versions of Python that Sage supports. The information regarding the supported versions can be found in the files build/pkgs/python3/spkg-configure.m4 and src/setup.cfg.m4.

Python 3.9 is the oldest supported version. Hence, all language and library features that are available in Python 3.9 can be used; but features introduced in Python 3.10 cannot be used. If a feature is deprecated in a newer supported version, it must be ensured that deprecation warnings issued by Python do not lead to failures in doctests.

Some key language and library features have been backported to older Python versions using one of two mechanisms:

  • from __future__ import annotations (see Python reference for __future__) modernizes type annotations according to PEP 563 (Postponed evaluation of annotations). All Sage library code that uses type annotations should include this __future__ import and follow PEP 563.

  • Backport packages

    The Sage library declares these packages as dependencies and ensures that versions that provide features of Python 3.11 are available.

Meta github issue #29756 keeps track of newer Python features and serves as a starting point for discussions on how to make use of them in the Sage library.


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.structure.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. Note that the .. skip line should not be included in your code; it is here to prevent doctests from running on this fake example.

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/src/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)
    <class '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 mix together the hash values of the components of the object that also play a part in comparison of objects by packing them into a tuple and hashing the tuple.

If a class does not define an __eq__() method it should not define a __hash__() operation either; if it defines __eq__() but not __hash__(), its instances will not be usable as items in hashable collections. If a class defines mutable objects and implements an __eq__() method, it should not implement __hash__(), since the implementation of hashable collections requires that a key’s hash value is immutable (if the object’s hash value changes, it will be in the wrong hash bucket).

See for more information on the subject.

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.

A method or a function accepts input described in the INPUT block of the docstring. If the input cannot be handled by the code, then it may raise an exception. The following aims to guide you in choosing from the most relevant exceptions to Sage. Raise

  • TypeError: if the input belongs to a class of objects that is not supported by the method. For example, a method works only with monic polynomials over a finite field, but a polynomial over rationals was given.

  • ValueError: if the input has a value not supported by the method. For example, the above method was given a non-monic polynomial.

  • ArithmeticError: if the method performs an arithmetic operation (sum, product, quotient, and the like) but the input is not appropriate.

  • ZeroDivisionError: if the method performs division but the input is zero. Note that for non-invertible input values, ArithmeticError is more appropriate. As derived from ArithmeticError, ZeroDivisionError can be caught as ArithmeticError.

  • NotImplementedError: if the input is for a feature not yet implemented by the method. Note that this exception is derived from RuntimeError.

If no specific error seems to apply for your situation, RuntimeError can be used. In all cases, the string associated with the exception should describe the details of what went wrong.

Integer return values#

Many functions and methods in Sage return integer values. Those should usually be returned as Sage integers of class Integer rather than as Python integers of class int, as users may want to explore the resulting integers’ number-theoretic properties such as prime factorization. Exceptions should be made when there are good reasons such as performance or compatibility with Python code, for instance in methods such as __hash__, __len__, and __int__.

To return a Python integer i as a Sage integer, use:

from sage.rings.integer import Integer
return Integer(i)

To return a Sage integer i as a Python ineger, use:

return int(i)


We mention two issues with importing: circular imports and importing large third-party modules. See also Dependencies and distribution packages for a discussion of imports from the viewpoint of modularization.

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.

If your module needs to make some precomputed data available at the top level, you can reduce its load time (and thus startup time, unless your module is imported using sage.misc.lazy_import) by using the decorator sage.misc.cachefunc.cached_function() instead. For example, replace

big_data = initialize_big_data()  # bad: runs at module load time


from sage.misc.cachefunc import cached_function

@cached_function                  # good: runs on first use
def big_data():
    return initialize_big_data()

Static typing#

Python libraries are increasingly annotated with static typing information; see the Python reference on typing.

For typechecking the Sage library, the project uses pyright; it automatically runs in the GitHub Actions CI and can also be run locally.

As of Sage 10.2, the Sage library only contains a minimal set of such type annotations. Pull requests that add more annotations are generally welcome.

The Sage library makes very extensive use of Cython (see chapter Coding in Cython). Although Cython source code often declares static types for the purpose of compilation to efficient machine code, this typing information is unfortunately not visible to static checkers such as Pyright. It is necessary to create type stub files (“.pyi”) that provide this information. Although various tools for writing and maintaining type stub files are available, creating stub files for Cython files involves manual work. There is hope that better tools become available soon, see for example cython/cython #5744. Contributing to the development and testing of such tools likely will have a greater impact than writing the typestub files manually.

For Cython modules of the Sage library, these type stub files would be placed next to the .pyx and .pxd files.

When importing from other Python libraries that do not provide sufficient typing information, it is possible to augment the library’s typing information for the purposes of typechecking the Sage library:

  • Create typestub files and place them in the directory SAGE_ROOT/src/typings. For example, the distribution pplpy provides the top-level package ppl, which publishes no typing information. We can create a typestub file SAGE_ROOT/src/typings/ppl.pyi or SAGE_ROOT/src/typings/ppl/__init__.pyi.

  • When these typestub files are working well, it is preferable from the viewpoint of the Sage project that they are “upstreamed”, i.e., contributed to the project that maintains the library. If a new version of the upstream library becomes available that provides the necessary typing information, we can update the package in the Sage distribution and remove the typestub files again from SAGE_ROOT/src/typings.

  • As a fallback, when neither adding typing annotations to source files nor adding typestub files is welcomed by the upstream project, it is possible to contribute typestubs files instead to the typeshed community project.


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 GitHub PR 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 a global namespace: this is when you want to remove a name from a global namespace (say, sage.all or some other file) but you want to keep the functionality available with an explicit import. This case is similar as the previous one: use a lazy import with deprecation. One detail: in this case, you don’t want the name lazy_import to be visible in the global namespace, so we add a leading underscore:

    from sage.misc.lazy_import import lazy_import as _lazy_import
    _lazy_import('sage.some.package', 'some_function', deprecation=666)
  • 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.