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
48383 Diophantine
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!
Quomodocumque 
There’s a criterion by J. Todd for arctangent identities (at least of a “standard” kind, of which the one you mention is an instance), and it has to do only with squares plus one, so squares minus cubes are probably unrelated…
See J. Todd, A problem on arc tangent relations, Amer. Math. Monthly 56 (1949) 517–528,
and
Click to access note-arctan.pdf
Well, when *I* was a “kid”, we had only the “Antwerp tables” from LNM 476 (proceedings of Modular Forms of One Variable, Antwerp 1975), which can now also be found online in W.Stein’s scans. Besides all modular elliptic curves of conductor up to 200 (Table 1), the tables contain other information – including all elliptic curves with conductor 2^f 3^g, or (almost) equivalently all primitive solutions of x^3 – y^2 = ± 2^a 3^b, attributing the list (Table 4a) to “the unpublished 1966 Manchester thesis of F.B. Coghlan”. Thus the rest is a decade older than the Rabinowitz paper you found. The solution corresponding to curve 1296G1 (which is remarkable for having a 21-isogeny) is #83:(x,y)=(5745,435447). This formulation of the problem allows a larger solution, #75: (x,y)=(8158,736844), corresponding to curve 20736P1. This is one of eight quadratic twists having the same conductor 20736=12^4, which is minimal for this (x,y); like 1296G1, these curves have a rational isogeny of unusual degree, here 13. For example, 20736P1 is 13-isogenous with the much simpler curve y^2=x^3+6x-8, corresponding to solution #24: (x,y)=(-2,4).
(Typo: the *list* is a decade older than Rabinowitz. It also contains Rabinowitz’s list as a proper subset.)
I should say that Rabinowitz’s paper has a note at the end saying “this is part of my thesis and after I submitted it I found out it was also in Coghlan’s thesis.”
Interesting. Meanwhile, the solution you mentioned turns out to have a quadratic twist of low enough conductor that it appears already in the “Antwerp” tables, at conductor 162. Those curves of conductor 162 are also mentioned in the “Remarks on Isogenies” in the same LNM volume, because they and their quadratic twists are the unique examples over Q of isogenies of degree 21: the curve X_0(21) is elliptic with 8 rational points, of which four are cusps and the other four correspond to this isogeny class.
I agree that this database is wonderful. “When I was a kid” the Antwerp volume hadn’t yet appeared, and you had to be lucky enough to have a professor who had access to privately circulating materials. Same with number theory packages like pari/gp — if you wanted to calculate a quadratic residue or a class number, you had to write your own subroutine. It’s
great that people took the time and trouble to pull all of this material onto one site.
The Serre/Tate correspondance (recently published by the French math. society) has a whole bunch of letters where they are enthusiastic about the capabilities of (if I remember right) their new HP 67s, or something like that. Tate, in particular, explains how he programs it to compute various things…
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