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

## Software packages useful for elliptic curve investigations

• ### 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 Fp 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

All the following are available, sometimes requiring the use of an add-on external program:
• 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.