This page was prepared for a workshop at the Workshop on Modular Forms: Arithmetic and Computation, BIRS (Banff, Canada) 3-8 June 2007.

Elliptic Curve Database

• Elliptic curve data for conductors up to 130000 (last major update 2006-09-24) -- and some supplementary data
  Interactive front-end to the data (for conductors up to 130000) by from Gonzalo Tornaria
• Tom Womack's lists
• Dujella's tables of High rank elliptic curves with prescribed torsion
• William Stein's modular forms database

Software packages useful for elliptic curve investigations

• mwrank and related programs:

  • mwrank home page
  
  

• Magma:

  Magma has a huge amount of functionality for working with elliptic curves over Q, number fields, finite fields, and general fields.

  It includes the database of all elliptic curves over Q with conductor less than 130000 (but none of the data in the database is made available apart from the curves themselves).

  
  

• Pari/GP:

  Pari/GP has a many functions for working with elliptic curves over Q, and a few fir general fields. There are also some add-on packages (e.g. mine for group structure over F_p and for analytic rank).

  An optional extra database has all elliptic curves of conductor up to 130000, together with ranks and generators mod torsion.

  
  

• SAGE:

  Has all Pari/GP functionality and access to mwrank and my other elliptic curve programs, but currently there is a lot not yet available (unless you also have Magma installed)

Elliptic Curve functionality

Elliptic Curves over Q

• basic invariants, conductors, torsion points
• point operations, heights and height pairing
• ranks, generators
• division polynomials
• L-series and analytic rank
• isogenies

Associated modular forms

• Magma can construct elliptic curves from (suitable!) modular forms, and vice versa.
• None of the other packages have (yet?).
• My programs for computing modular elliptic curves are not suitable for general use...

Elliptic Curves over other fields

• Over finite fields: Magma can do everything (order, structure, discrete log, Weil pairing). Pari/gp can by default only find group order but add-ons can do the rest. SAGE?
• Over local fields: Magma, Pari/gp and SAGE can find Tate parameters. mwrank has some functionality for working in component groups.
• Number fields other than Q: Magma can find conductors, local (and global when class number=1) minimal models, heights of points, and ranks via 2-descent.