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.
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! 
David Lowry-Duda August 7, 2020 at 4:15 pm on AboutThat's great! If there are particular plots that you're interested in, let me know. But I (very recently) posted a library for making the plots https://davidlowryduda.com/phase_mag_plot-a-sage-package-for-plotting-complex-functions/ Alternately, I make sage/python notebooks available for many of my visualizations. These are usually linked to within the post. If you see something that you want to replicate and find the code, leave a ---
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 
Peter Humphries October 7, 2019 at 4:03 am on Notes from a talk at the Maine-Quebec Number Theory ConferenceOmega results of this form using this method are presented quite nicely in Chapter 15 of Montgomery and Vaughan; they give applications towards sign changes of the Chebyshev psi function and of partial sums of the Mobius function. 
David Lowry-Duda June 16, 2019 at 1:55 am on Choosing functions and generating figures for “When are there continuous choices for the mean value abscissa?”At first, we chose 2 because it was the other "obvious" point that we hadn't yet specified. We already controlled the values at 0, 1, and 3. But I later realized it doesn't matter what point we choose. What we're really doing is adding one extra degree of freedom and using it to minimize the L2 norm of the derivative ---
PC June 16, 2019 at 1:14 am on Choosing functions and generating figures for “When are there continuous choices for the mean value abscissa?”Why did you choose 2 for the pt in the L2 part? What if you chose 1.5 or something? 
Donna ebert September 27, 2018 at 3:55 pm on AboutI appreciate your taking the time to reply. The version of FreeCell I play online does count taking a card from the tableau (and the Freecells) as a move when adding them to the foundation. I never realized others did not.