Online Toolbox For Number Theorists

Sometimes I wonder: how did people do research before the internet? Not only do electronic communication and dropbox folders make long-distance collaborations a piece of cake, there are also just so many useful things on the internet. When I stand back and take stock of my number theory research toolbox there are a few online items that improve my research life beyond measure. I want to share those with you.

To begin, the metaphoric Phillips head screwdriver in my online toolbox is a highly specified and but oft-used resource: the number field database. This database was compiled and maintained by John Jones and David Roberts, and a few other contributors — both human and software. It is basically a tool that takes in a list of field parameters and spits out a list of number fields satisfying every item on the list. Let’s say, for example, that I’m trying to get my hands on every imaginary degree 2 number field with discriminant bounded by 100, and just for fun, let’s say it can only ramify at 3 and 5. You plug those parameters into the database, et voilà, it returns the number field with minimal polynomial x²-x+4.

And as if the database on its own wasn’t cool enough, you can actually load it into Sage and do number field computations to your heart’s content. Speaking of Sage, if the number field database is the Phillips head in my toolbox, the Sagecloud is the hammer.

Even the most amateur toolbox isn’t complete without a reasonable selection of flat-head screwdrivers, and the L-functions and Modular Forms Database (LMFDB) is exactly that. The LMFDB can do a lot of things. In particular it hosts the number field database mentioned above, and then some. It is an ever growing library of tables, formulas, links, and references for L-functions, modular forms, and related objects. Generally, L-functions are just the analytic continuation of a Dirichlet series which have an associated Euler product. Examples of L-fucntions come out of modular forms, Diriclet characters and elliptic curves, but perhaps the most famous example is the Riemann zeta function.

A search of the LMFDB for elliptic curves with conductor 389 returns a list of curves with some easy-to-read essential invariants.

The LMFDB has a similar setup to the number field database, in that you can ask for an object with the prescribed characteristics and you get back a full list of candidates. But now the search has broadened. Say, for example, I want to see every possible elliptic curve over the rationals with conductor 389. The LMFDB returns a list complete with each curve’s rank, Weierstrass coefficients, j-invariant, and index in the Cremona elliptic curve database.

Finally, every toolbox has that one item (for me it’s my level) that I’ve never actually used, but it’s still fun to take out and play with from time to time. And that, for me, is the Online Encyclopedia of Integer Sequences. This is not to say that it’s not a useful tool, I know plenty of people who use it in their research, but I’ve never had occasion to use it beyond blissfully throwing in random strings of numbers and learning about new sequences. Math writer Alex Bellos wrote a very nice piece about the OEIS for The Guardian in honor of its 50th birthday.

These are just a few tools for the number theoretic among us, and surely there are many more. Let us know if there are other online tools and databases that help you in your research. And if you know how people did research before the internet, tweet it at me @extremefriday.