Function Field Database

An optional database of function fields may be downloaded from the Magma website. This section defines the interface to that database. Each database is associated to a given finite field and extension degree. The supported combinations are:

\({\mathbb{F}}_2\):

degrees 2, 3

\({\mathbb{F}}_3\):

degree 2

\({\mathbb{F}}_4\):

degrees 2, 3, 5

\({\mathbb{F}}_5\):

degrees 2, 3, 4, 8

\({\mathbb{F}}_7\):

degree 9

\({\mathbb{F}}_{11}\):

degree 3

\({\mathbb{F}}_{13}\):

degree 3

For each function field in the database, the following information is stored and may be used to limit the function fields of interest via the sub< D | : parameters> constructor: The genus; the number of degree one places; the class number; and the class group.

Creation

FunctionFieldDatabase(q, d): RngIntElt, RngIntElt -> DB

Returns a database object for the function fields of degree \(d\) over \({\mathbb{F}}_q\).

sub< D | : parameters>: DB -> DB

Genus                  : RngIntElt                    Default: 
NumberOfDegreeOnePlaces: RngIntElt                    Default: 

Returns a sub-database of \(D\), restricting (or further restricting, if \(D\) is already a sub-database of the full database) the contents to those function fields satisfying the specified conditions. Note that it is not possible to “undo” restrictions with this constructor — the results are always at least as limited as \(D\) is.

The parameter Genus may be used to restrict the search to fields with the specified genus.

The parameter NumberOfDegreeOnePlaces may be used to restrict the search to only those fields with the specified number of places of degree one.

Access

BaseField(D): DB -> FldFin

CoefficientField(D): DB -> FldFin

Returns the finite field underlying each function field in the database.

Degree(D): DB -> RngIntElt

Returns the degree of each function field in the database.

# D: DB -> RngIntElt

NumberOfFields(D): DB -> RngIntElt

Returns the number of function fields stored in the database.

FunctionFields(D): DB -> [ FldFunG ]

Returns the sequence of function fields stored in the database.

Example: Genus Distribution in the Function Field Database (ex-f1c514)

The genus of a degree four function field is at most 6. We can see the distribution in a database by counting the size of appropriate sub-databases:

> D := FunctionFieldDatabase(5, 4);
> #D;
196380
> [ #sub<D |: Genus := g> : g in [0..6] ];
[ 60, 480, 960, 12960, 35040, 63120, 83760 ]

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