The elliptic curves over $\Q$ in the LMFDB come from three sources:

- Complete lists of elliptic curves defined over $\Q$ for a given conductor $N$ were computed for $N\le500000$ by John Cremona using the modular symbols method described in [Cremona97, MR:1628193], as implemented in his eclib package [10.5281/zenodo.29671].
- For conductors only divisible by primes $p\le7$, complete lists of curves were computed by Benjamin Matschke and Rafael von Känel by solving appropriate Mordell equations [arXiv:1605.06079], and independently by Michael Bennett, Adela Gherga, and Andrew Rechnitzer [BGR2019].
- For prime conductors, extensive but incomplete lists of curves were computed by William Stein and Mark Watkins as part of the Stein-Watkins database [SW2002]. Independently, complete lists were computed by Bennett, Gherga and Rechnitzer [BGR2019], both extending the range of the Stein-Watkins table and including curves omitted there.

The additional data for each curve and isogeny class was also computed by John Cremona using a combination of eclib, Magma, SageMath, and Pari/GP unless otherwise specified.

- Isogeny classes were computed using with eclib or Sage, which uses methods of Cremona, Tsukazaki and Watkins to compute isogenies of prime degree.
- The ranks and generators were mostly computed using mwrank (part of eclib), with larger generators of rank 1 curves computed using either Pari/GP's ellheegner function (implemented by Bill Allombert using his enhancements of the algorithm developed by John Cremona, Christophe Delaunay and Mark Watkins) or Magma's HeegnerPoint function (implemented by Steve Donnelly and Mark Watkins, based on the same method).
- Integral points were computed using both the Sage implementation of the method based on bounds on elliptic logarithms and LLL-reduction, implemented by John Cremona with Michael Mardaus and Tobias Nagell, and also the Magma implementation of Steve Donnelly (which uses some additional unpublished ideas)
- Modular degrees were computed using Mark Watkins's sympow library via Sage.
- Special values of the L-function were computed by Pari/GP via Sage.
- Information on mod-$\ell$ Galois representations was computed by Andrew Sutherland using the methods of [10.1017/fms.2015.33, arXiv:1504.07618] (including a generalization to handle curves with complex multiplication).
- Information on 2-adic Galois representations was computed using Jeremy Rouse's Magma implementation of the algorithm of Rouse and Zureick-Brown [10.1007/s40993-015-0013-7, arXiv:1402.5997].
- Information on $\ell$-adic Galois representations was computed using the algorithm of Rouse, Sutherland, and Zureick-Brown [arXiv:2106.11141]; a Magma implementation is available in the associated GitHub repository.
- The graphs of the real locus were computed using David Roe's function implemented in Sage.
- Iwasawa invariants were computed by Robert Pollack.
- Torsion growth data was computed by Enrique González Jiménez and Filip Najman.

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by Benjamin Matschke on 2022-01-13 10:02:58

**Referred to by:**

**History:**(expand/hide all)

- 2022-11-10 18:27:59 by Andrew Sutherland
- 2022-11-10 18:27:32 by Andrew Sutherland
- 2022-11-10 18:19:28 by Andrew Sutherland
- 2022-01-13 10:02:58 by Benjamin Matschke (Reviewed)
- 2021-07-19 11:45:19 by Andrew Sutherland (Reviewed)
- 2021-04-17 09:06:08 by Andrew Sutherland (Reviewed)

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