Very happy to see that the L-functions and Modular Forms Database is now live!
When I was a kid we looked up our elliptic curves in Cremona’s tables, on paper. Then William Stein created the Modular Forms Database (you can still go there but it doesn’t really work) and suddenly you could look at the q-expansions of cusp forms in whatever weight and level you wanted, up to the limits of what William had computed.
The LMFDB is a sort of massively souped up version of Cremona and Stein, put together by a team of dozens and dozens of number theorists, including too many friends of mine to name individually. And it’s a lot more than what the title suggests: the incredibly useful Jones-Roberts database of local fields is built in; there’s a database of genus 2 curves over Q with small conductor; it even has Maass forms! I’ve been playing with it all night. It’s like an adventure playground for number theorists.
One of my first trips through Stein’s database came when I was a postdoc and was thinking about Ljunggren’s equation y^2 + 1 = 2x^4. This equation has a large solution, (13,239), which has to do with the classical identity
It turns out, as I explain in an old survey paper, that the existence of such a large solution is “explained” by the presence of a certain weight-2 cuspform in level 1024 whose mod-5 Galois representation is reducible.
With the LMFDB, you can easily wander around looking for more such examples! For instance, you can very easily ask the database for non-CM elliptic curves whose mod-7 Galois representation is nonsurjective. Among those, you can find this handsome curve of conductor 1296, which has very large height relative to its conductor. Applying the usual Frey curve trick you can turn this curve into the Diophantine oddity
3*48383^2 – (1915)^3 = 2^13.
Huh — I wonder whether people ever thought about this Diophantine problem, when can the difference between a cube and three times a square be a power of 2? Of course they did! I just Googled
and found this paper of Stanley Rabinowitz from 1978, which finds all solutions to that problem, of which this one is the largest.
Now whether you can massage this into an arctan identity, that I don’t know!