Sage (see http://sagemath.org) is a comprehensive mathematical software system for computations in many areas of pure and applied mathematics. We program Sage using the mainstream programming language Python (see http://python.org), or its compiled variant Cython. It is also very easy to efficiently use code written in C/C++ from Sage.

The author of this article started the Sage project in 2005.

Sage is free and open source, meaning you can change any part of Sage and redistribute the result without having to pay any license fees, and Sage can also leverage the power of commercial mathematical software such as Magma and Mathematica, if you happen to have access to those closed source commercial systems.

This document assumes no prior knowledge of either Python or Sage. Our goal is to help number theorists do computations involving number fields and modular forms using Sage.

TODO: Overview of Article

As you read this article, please try every example in Sage, and make sure things works as I claim, and do all of the exercises. Moreover, you should experiment by typing in similar examples and checking that the output you get agrees with what you expect.

To use Sage, install it on your computer, and use either the command
line or start the Sage notebook by typing `notebook()` at the
command line.

We show Sage sessions as follows:

```
sage: factor(123456)
2^6 * 3 * 643
```

This means that if you type `factor(123456)` as input to Sage, then
you’ll get `2^6 * 3 * 643` as output. If you’re using the Sage
command line, you type `factor(123456)` and press enter; if you’re
using the Sage notebook via your web browser, you type
`factor(123456)` into an input cell and press shift-enter; in the
output cell you’ll see `2^6 * 3 * 643`.

After trying the `factor` command in the previous
paragraph (do this now!), you should try factoring some other
numbers.

Note

What happens if you factor a negative number? a rational number?

You can also draw both 2d and 3d pictures using Sage. For example, the following input plots the number of prime divisors of each positive integer up to \(500\).

```
sage: line([(n, len(factor(n))) for n in [1..500]])
Graphics object consisting of 1 graphics primitive
```

And, this example draws a similar 3d plot:

```
sage: v = [[len(factor(n*m)) for n in [1..15]] for m in [1..15]]
sage: list_plot3d(v, interpolation_type='nn')
Graphics3d Object
```

*The main difference between Sage and Pari is that Sage is vastly larger than Pari with a much wider range of functionality, and has many more data types and much more structured objects.*Sage in fact includes Pari, and a typical Sage install takes nearly a gigabyte of disk space, whereas a typical Pari install is much more nimble, using only a few megabytes. There are many number-theoretic algorithms that are included in Sage, which have never been implemented in Pari, and Sage has 2d and 3d graphics which can be helpful for visualizing number theoretic ideas, and a graphical user interface. Both Pari and Sage are free and open source, which means anybody can read or change anything in either program, and the software is free.*The biggest difference between Sage and Magma is that Magma is closed source, not free, and difficult for users to extend.*This means that most of Magma cannot be changed except by the core Magma developers, since Magma itself is well over two million lines of compiled C code, combined with about a half million lines of interpreted Magma code (that anybody can read and modify). In designing Sage, we carried over some of the excellent design ideas from Magma, such as the parent, element, category hierarchy.*Any mathematician who is serious about doing extensive computational work in algebraic number theory and arithmetic geometry is strongly urged to become familiar with all three systems*, since they all have their pros and cons. Pari is sleek and small, Magma has much unique functionality for computations in arithmetic geometry, and Sage has a wide range of functionality in most areas of mathematics, a large developer community, and much unique new code.