The Stein-Watkins table of elliptic curves.

Sage gives access to the Stein-Watkins table of elliptic curves, via an optional package that you must install. This is a huge database of elliptic curves. You can download the database as a 2.6GB Sage package from http://modular.ucsd.edu/sagedb/, which you install with the command

sage -i stein-watkins-ecdb.spkg

You can also download a small version, without having to explicitly download anything from a website, using the command

sage -i stein-watkins-ecdb-mini

This database covers a wide range of conductors, but unlike CremonaDatabase(), this database need not list all curves of a given conductor. It lists the curves whose coefficients aren’t “too large” (see [Stein-Watkins, Ants 5]).

• The command SteinWatkinsAllData(n) returns an iterator over the curves in the Stein-Watkins table, which contains elliptic curves of conductor between and . Here can be between and , inclusive.
• The command SteinWatkinsPrimeData(n) returns an iterator over the curves in the Stein-Watkins table, which contains prime conductor elliptic curves of conductor between and . Here varies between and , inclusive.

EXAMPLES: We obtain the first table of elliptic curves.

sage: d = SteinWatkinsAllData(0)
sage: d
Stein-Watkins Database a.0 Iterator

We type d.next() to get each isogeny class of curves from d:

sage: C = d.next()                                   # optional - stein_watkins_database
sage: C                                              # optional - stein_watkins_database
Stein-Watkins isogeny class of conductor 11
sage: d.next()                                       # optional - stein_watkins_database
Stein-Watkins isogeny class of conductor 14
sage: d.next()                                       # optional - stein_watkins_database
Stein-Watkins isogeny class of conductor 15

An isogeny class has a number of attributes that give data about the isogeny class, such as the rank, equations of curves, conductor, leading coefficient of -function, etc.

sage: C.data                                         # optional - stein_watkins_database
['11', '[11]', '0', '0.253842', '25', '+*1']
sage: C.curves                                       # optional - stein_watkins_database
[[[0, -1, 1, 0, 0], '(1)', '1', '5'],
 [[0, -1, 1, -10, -20], '(5)', '1', '5'],
 [[0, -1, 1, -7820, -263580], '(1)', '1', '1']]
sage: C.conductor                                    # optional - stein_watkins_database
11
sage: C.leading_coefficient                          # optional - stein_watkins_database
'0.253842'
sage: C.modular_degree                               # optional - stein_watkins_database
'+*1'
sage: C.rank                                         # optional - stein_watkins_database
0
sage: C.isogeny_number                               # optional - stein_watkins_database
'25'

If we were to continue typing d.next() we would iterate over all curves in the Stein-Watkins database up to conductor . We could also type for C in d: ...

To access the data file starting at do the following:

sage: d = SteinWatkinsAllData(1)                    # optional - stein_watkins_database
sage: C = d.next()                                  # optional - stein_watkins_database
sage: C                                             # optional - stein_watkins_database
Stein-Watkins isogeny class of conductor 100002
sage: C.curves                                      # optional - stein_watkins_database
[[[1, 1, 0, 112, 0], '(8,1,2,1)', 'X', '2'],
 [[1, 1, 0, -448, -560], '[4,2,1,2]', 'X', '2']]

Next we access the prime-conductor data:

sage: d = SteinWatkinsPrimeData(0)                 # optional - stein_watkins_database
sage: C = d.next()                                 # optional - stein_watkins_database
sage: C                                            # optional - stein_watkins_database
Stein-Watkins isogeny class of conductor 11

Each call d.next() gives another elliptic curve of prime conductor:

sage: C = d.next()                                 # optional - stein_watkins_database
sage: C                                            # optional - stein_watkins_database
Stein-Watkins isogeny class of conductor 17
sage: C.curves                                     # optional - stein_watkins_database
[[[1, -1, 1, -1, 0], '[1]', '1', '4'],
 [[1, -1, 1, -6, -4], '[2]', '1', '2x'],
 [[1, -1, 1, -1, -14], '(4)', '1', '4'],
 [[1, -1, 1, -91, -310], '[1]', '1', '2']]
sage: C = d.next()                                 # optional - stein_watkins_database
sage: C                                            # optional - stein_watkins_database
Stein-Watkins isogeny class of conductor 19

class sage.databases.stein_watkins.SteinWatkinsAllData(num)

Class for iterating through one of the Stein-Watkins database files for all conductors.

iter_levels()

Iterate through the curve classes, but grouped into lists by level.

next()

class sage.databases.stein_watkins.SteinWatkinsIsogenyClass(conductor)

class sage.databases.stein_watkins.SteinWatkinsPrimeData(num)

Bases: sage.databases.stein_watkins.SteinWatkinsAllData

sage.databases.stein_watkins.ecdb_num_curves(max_level=200000)