Sage Days 87 Demo: Interfacing between sage and the LMFDB
Interfacing sage and the LMFDB — a prototype¶
The lmfdb and sagemath are both great things, but they don’t currently talk to each other. Much of the lmfdb calls sage, but the lmfdb also includes vast amounts of data on $L$-functions and modular forms (hence the name) that is not accessible from within sage.
This is an example prototype of an interface to the lmfdb from sage. Keep in mind that this is a prototype and every aspect can change. But we hope to show what may be possible in the future. If you have requests, comments, or questions, please request/comment/ask either now, or at my email: david@lowryduda.com.
Note that this notebook is available on http://davidlowryduda.com or https://gist.github.com/davidlowryduda/deb1f88cc60b6e1243df8dd8f4601cde, and the code is available at https://github.com/davidlowryduda/sage2lmfdb
Let’s dive into an example.
# These names will change
from sage.all import *
import LMFDB2sage.elliptic_curves as lmfdb_ecurve
lmfdb_ecurve.search(rank=1)
Es = lmfdb_ecurve.search(rank=1)
E = Es[0]
print(type(E))
th
# Try autocompleting the following. It has all the things!
print(dir(E))
This gives quick access to some data that is not stored within the LMFDB, but which is relatively quickly computable. For example,
E.defining_ideal()
Es = lmfdb_ecurve.search(conductor=11050, torsion_order=2)
print("There are {} curves returned.".format(len(Es)))
E = Es[0]
print(E)
print(E.gens())
print(E.rank())
print(E.regulator())
res = []
%time for E in Es: res.append(E.gens()); res.append(E.rank()); res.append(E.regulator())
search() function.In this case, elliptic curves over the rationals are only an okay example, as they’re really well studied and sage can compute much of the data very quickly. On the other hand, through the LMFDB there are millions of examples and corresponding data at one’s fingertips.
This is where we’re really looking for input.¶
Think of what you might want to have easy access to through an interface from sage to the LMFDB, and tell us. We’re actively seeking comments, suggestions, and requests. Elliptic curves over the rationals are a prototype, and the LMFDB has lots of (much more challenging to compute) data. There is data on the LMFDB that is simply not accessible from within sage.
email: david@lowryduda.com, or post an issue on https://github.com/LMFDB/lmfdb/issues
Now let’s describe what’s going on under the hood a little bit¶
There is an API for the LMFDB at http://beta.lmfdb.org/api/. This API is a bit green, and we will change certain aspects of it to behave better in the future. A call to the API looks like
http://beta.lmfdb.org/api/elliptic_curves/curves/?rank=i1&conductor=i11050
The result is a large mess of data, which can be exported as json and parsed.
But that’s hard, and the resulting data are not sage objects. They are just strings or ints, and these require time and thought to parse.
So we created a module in sage that writes the API call and parses the output back into sage objects. The 22 curves given by the above API call are the same 22 curves returned by this call:
Es = lmfdb_ecurve.search(rank=1, conductor=11050, max_items=25)
print(len(Es))
E = Es[0]
# Execute this cell for the documentation
print(lmfdb_ecurve.search.__doc__)
# So, for instance, one could perform the following search, finding a unique elliptic curve
lmfdb_ecurve.search(rank=2, torsion_order=3, degree=4608)
What if there are no curves?¶
If there are no curves satisfying the search criteria, then a message is displayed and that’s that. These searches may take a couple of seconds to complete.
For example, no elliptic curve in the database has rank 5.
lmfdb_ecurve.search(rank=5)
How does one step through the data?¶
Right now, at most 100 curves are returned in a single API call. This is the limit even from directly querying the API. But one can pass in the argument base_item (the name will probably change… to skip? or perhaps to offset?) to start returning at the base_itemth element.
from pprint import pprint
pprint(lmfdb_ecurve.search(rank=1, max_items=3)) # The last item in this list
print('')
pprint(lmfdb_ecurve.search(rank=1, max_items=3, base_item=2)) # should be the first item in this list
max_conductor or min_conductor (or arguments of that type). But it will sometime. (This introduces a few extra difficulties on the server side, and so it will take some extra time to decide how to do this).
lmfdb_ecurve.search(rank=1, min_conductor=500, max_conductor=10000) # Not implemented
EllipticCurve_rational_field_lmfdb class constructs a sage elliptic curve from the json and overrides (somem of the) the default methods in sage if there is quicker data available on the LMFDB. In principle, this new object is just a sage object with some slightly different methods.Generically, documentation and introspection on objects from this class should work. Much of sage’s documentation carries through directly.
print(E.gens.__doc__)
regulator().)
print(E.regulator.__doc__)
This concludes our demo of an interface between sage and the LMFDB.¶
Thank you, and if you have any questions, comments, or concerns, please find me/email me/raise an issue on LMFDB’s github.
Yavor Kirov April 14, 2021 at 3:24 am on Long live opalstack!Thabk you! I was looking for an opinion of a long-term user of Opalstack so your answer was exactly what I needed. 
David Lowry-Duda April 13, 2021 at 9:32 pm on Long live opalstack!Hi Yavor, I'm very happy with Opalstack. After the initial transition, there have been almost no problems. I'll note that for a few months, opalstack's email setup was rocky, but they figured that out. I'd recommend it. They've been pretty responsive to problems, too. You can see what's troubling people now on their community/support forum: https://community.opalstack.com/. I know a somewhat ---
Tomek March 18, 2021 at 5:09 pm on A response to FtYoU’s question on RedditGod bless you David. Great explanation! 
Jan van Delden February 25, 2021 at 5:07 pm on phase_mag_plot: a sage package for plotting complex functionsI took the liberty to download your SageMath module and changed a few options. Since I was not completely clear on the interaction between lightness and color (defined by the argument of the function to be displayed) I decided to weigh these differently. Color, from the colorwheel by 0.8 and lightness by 0.2. I computed the color first and weighed ---
eliot December 23, 2020 at 4:07 pm on Trigonometric and related substitutions in integralsFor Euler substitutions, do you mind explaining the reasoning for the t term in x*sqrt(a) + t. Is there anywhere else I can read up on this? Thanks for this post! 
Nguyen Xuan Son April 11, 2020 at 4:53 pm on AboutHi Could you please share the python code to generating beautiful plots? I plan to use it for my son to get interested in python programming. Son