Elliptic curves: Magma versus Sage

Elliptic Curves

Elliptic curves are certain types of nonsingular plane cubic curves, e.g., y^2 = x^3 + ax +b, which are central to both number theory and cryptography (e.g., they are used to compute the hash in bitcoin).

Magma and Sage

If you want to do a wide range of explicit computations with elliptic curves, for research purposes, you will very likely use SageMath or Magma. If you're really serious, you'll use both.

Both Sage and Magma are far ahead of all other software (e.g., Mathematica, Maple and Matlab) for elliptic curves.

• Magma reference manual about elliptic curves.
• Sage reference manual about elliptic curves.
• Pari reference manual about elliptic curves. -- pari is part of Sage and has some unique powerful functionality, e.g., ellheegner...

A Little History

When I started contributing to Magma in 1999, I remember that Magma was way, way behind Pari. I remember having lunch with John Cannon (founder of Magma), and telling him I would no longer have to use Pari if only Magma would have dramatically faster code for computing point counts on elliptic curves.

A few years later, John wisely hired Mark Watkins to work fulltime on Magma, and Mark has been working there for over a decade. Mark is definitely one of the top people in the world at implementing (and using) computational number theory algorithms, and he's ensured that Magma can do a lot. Some of that "do a lot" means catching up with (and surpassing!) what was in Pari and Sage for a long time (e.g., point counting, p-adic L-functions, etc.)

However, in addition, many people have visited Sydney and added extremely deep functionality for doing higher descents to Magma, which is not available in any open source software. Search for Magma in this paper to see how, even today, there seems to be no open source way to compute the rank of the curve y² = x³ + 169304x + 25788938.  (The rank is 0.)

Two Codebases

There are several elliptic curves algorithms available only in Magma (e.g., higher descents) ... and some available only in Sage (L-function rank bounds, some overconvergent modular symbols, zeros of L-functions, images of Galois representations). I could be wrong about functionality not being in Magma, since almost anything can get implemented in a year...

The code bases are almost completely separate, which is a very good thing. Any time something gets implemented in one, it gets (or should get) tested via a big run on elliptic curves up to some bound in the other. This sometimes results in bugs being found. I remember refereeing the "integral points" code in Sage by running it against all curves up to some bound and comparing to what Magma output, and getting many discrepancies, which showed that there were bugs in both Sage and Magma.
Thus we would be way better off if Sage could do everything Magma does (and vice versa).

"If you were new faculty, would you start something like SageMathCloud sooner?"

I was recently asked by a young academic: "If you were a new faculty member again, would you start something like SageMathCloud sooner or simply leave for industry?" The academic goes on to say "I am increasingly frustrated by continual evidence that it is more valuable to publish a litany of computational papers with no source code than to do the thankless task of developing a niche open source library; deep mathematical software is not appreciated by either mathematicians or the public."

I wanted to answer that "things have gotten better" since back in 2000 when I started as an academic who does computation. Unfortunately, I think they have gotten worse. I do not understand why. In fact, this evening I just received the most recent in a long string of rejections by the NSF.

Regarding a company versus taking a job in industry, for me personally there is no point in starting a company unless you have a goal that can only be accomplished via a company, since building a business from scratch is extremely hard and has little to do with math or research. I do have such a goal: "create a viable open source alternative to Mathematica, etc...". I was very clearly told by Michael Monagan (co-founder of Maplesoft) in 2006 that this goal could not be accomplished in academia, and I spent the last 10 years trying to prove him wrong.

On the other hand, leaving for a job in industry means that your focus will switch from "pure" research to solving concrete problems that make products better for customers. That said, many of the mathematicians who work on open source math software do so because they care so much about making the experience of using math software much better for the math community. What often drives Sage developers is exactly the sort of passionate care for "consumer focus" and products that also makes one successful in industry. I'm sure you know exactly what I mean, since it probably partly motivates your work. It is sad that the math community turns its back on such people. If the community were to systematically embrace them, instead of losing all these $300K+/year engineers to mathematics entirely -- which is exactly what we do constantly -- the experience of doing mathematics could be massively improved into the future. But that is not what the community has chosen to do. We are shooting ourselves in the foot.

Now that I have seen how academia works from the inside over 15 years I'm starting to understand a little why these things change very slowly, if ever. In the mathematics department I'm at, there are a small handful of research areas in pure math, and due to how hiring works (voting system, culture, etc.) we have spent the last 10 years hiring in those areas little by little (to replace people who die/retire/leave). I imagine most mathematics departments are very similar. "Open source software" is not one of those traditional areas. Nobody will win a Fields Medal in it.

Overall, the mathematical community does not value open source mathematical software in proportion to its value, and doesn't understand its importance to mathematical research and education. I would like to say that things have got a lot better over the last decade, but I don't think they have. My personal experience is that much of the "next generation" of mathematicians who would have changed how the math community approaches open source software are now in industry, or soon will be, and hence they have no impact on academic mathematical culture. Every one of my Ph.D. students are now at Google/Facebook/etc.

We as a community overall would be better off if, when considering how we build departments, we put "mathematical software writers" on an equal footing with "algebraic geometers". We should systematically consider quality open source software contributions on a potentially equal footing with publications in journals.

To answer the original question, YES, knowing what I know now, I really wish I had started something like SageMathCloud sooner. In fact, here's the previously private discussion from eight years ago when I almost did.

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- There is a community generated followup ...

- Relevant blog post: About Software Development in Companies, Communities and the Academia

Open source is now ready to compete with Mathematica for use in the classroom

When I think about what makes SageMath different, one of the most fundamental things is that it was created by people who use it every day.  It was created by people doing research math, by people teaching math at universities, and by computer programmers and engineers using it for research.  It was created by people who really understand computational problems because we live them.  We understand the needs of math research, teaching courses, and managing an open source project that users can contribute to and customize to work for their own unique needs.

The tools we were using, like Mathematica, are clunky, very expensive, and just don't do everything we need.  And worst of all, they are closed source software, meaning that you can't even see how they work, and can't modify them to do what you really need.  For teaching math, professors get bogged down scheduling computer labs and arranging for their students to buy and install expensive software.

So I started SageMath as an open source project at Harvard in 2004, to solve the problem that other math software is expensive, closed source, and limited in functionality, and to create a powerful tool for the students in my classes.  It wasn't a project that was intended initially as something to be used by hundred of thousands of people.  But as I got into the project and as more professors and students started contributing to the project, I could clearly see that these weren't just problems that pissed me off, they were problems that made everyone angry.

The scope of SageMath rapidly expanded.  Our mission evolved to create a free open source serious competitor to Mathematica and similar closed software that the mathematics community was collective spending hundreds of millions of dollars on every year. After a decade of work by over 500 contributors, we made huge progress.

But installing SageMath was more difficult than ever.  It was at that point that I decided I needed to do something so that this groundbreaking software that people desperately needed could be shared with the world.

So I created SageMathCloud, which is an extremely powerful web-based collaborative way for people to easily use SageMath and other open source software such as LaTeX, R, and Jupyter notebooks easily in their teaching  and research.   I created SageMathCloud based on nearly two decades of experience using math software in the classroom and online, at Harvard, UC San Diego, and University of Washington.

SageMathCloud is commercial grade, hosted in Google's cloud, and very large classes are using it heavily right now.  It solves the installation problem by avoiding it altogether.  It is entirely open source.

Open source is now ready to directly compete with Mathematica for use in the classroom.  They told us we could never make something good enough for mass adoption, but we have made something even better.  For the first time, we're making it possible for you to easily use Python and R in your teaching instead of Mathematica; these are industry standard mainstream open source programming languages with strong support from Google, Microsoft and other industry leaders.   For the first time, we're making it possible for you to collaborate in real time and manage your course online using the same cutting edge software used by elite mathematicians at the best universities in the world.

A huge community in academia and in industry are all working together to make open source math software better at a breathtaking pace, and the traditional closed development model just can't keep up.

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