GAP (QaoS) - Chapter 3: User Functions

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3 User Functions
3.1 Retrieve objects matching a query
3.1-1 QaosTransitiveGroup
3.1-2 QaosNumberField
3.1-3 QaosDefaultLimit
3.2 Examine the contents of a database collection
3.2-1 QaosResult
3.2-2 QaosResult
3.2-3 AsGroup
3.2-4 AsField

### 3 User Functions

QaoS provides some user functions to obtain information from the databases. Currently, we support queries for transitive groups and algebraic number fields. The corresponding commands are `QaosTransitiveGroup` (3.1-1) and `QaosNumberField` (3.1-2).

#### 3.1 Retrieve objects matching a query

##### 3.1-1 QaosTransitiveGroup

`‣ QaosTransitiveGroup`( query[, optarg] ) | ( function )

Return transitive groups matching query. The amount of results is limited by the global variable `QaosDefaultLimit` (3.1-3). Optional argument optarg is a record with some of the following components:

- `Action` (string)

  - determine an action to perform on query

  - default value: "query"

  - possible values: "query", "count"

- `ColGroups` (list of strings)

  - determine which result groups should be returned; results of QaoS are usually grouped by a so-called `column group'. In the web interface these column groups are choosable by checking the corresponding name under the text field for the query.

  - default value: ["size","props","perm-struct"]

  - possible values: combinations of "size", "props", "perm-struct", "fp-struct"

- `Limit` (integer/string)

  - determine how many results are retrieved (maximally)

  - default value: `QaosDefaultLimit` (3.1-3)

  - possible values: any positive integer or "all"

- `Offset` (integer/string)

  - determine an offset on the set of results, this may be used to sequentially retrieve blocks of results.

  - default value: 0

  - possible values: any non-negative integer

##### 3.1-2 QaosNumberField

`‣ QaosNumberField`( query[, optarg] ) | ( function )

Return number fields matching query. The amount of results is limited by the global variable `QaosDefaultLimit` (3.1-3). Optional argument optarg is a record with some of the following components:

- `Action` (string)

  - determine an action to perform on query

  - default value: "query"

  - possible values: "query", "count"

- `ColGroups` (list of strings)

  - determine which result groups should be returned; results of QaoS are usually grouped by a so-called `column group'. In the web interface these column groups are choosable by checking the corresponding name under the text field for the query.

  - default value: ["size","struct","clsgroup","galgroup"]

  - possible values: combinations of "size", "struct", "clsgroup", "galgroup", "galprops"

- `Limit` (integer/string)

  - determine how many results are retrieved (maximally)

  - default value: `QaosDefaultLimit` (3.1-3)

  - possible values: any positive integer or "all"

- `Offset` (integer/string)

  - determine an offset on the set of results, this may be used to sequentially retrieve blocks of results.

  - default value: 0

  - possible values: any non-negative integer

##### 3.1-3 QaosDefaultLimit

`‣ QaosDefaultLimit` | ( global variable )

An Integer or String. Determine the (maximal) amount of results returned in a query. Usually, this can be overridden with the `Limit`-component in optargs. The default value is 25. The value can be any positive integer or the string "all", in which case no limit is set and all results are returned. CAUTION: Setting this variable to "all" may result in exhaustive use of memory, network bandwidth, and time. Therefore, always consider to perform the count action on a query before loading all of the results. The database of number fields contains more than 1.35 million number field objects. Downloading them all means retrieving a string of approx. 1.8 GB length!!!

The following examples show their usage: Query for transitive groups of degree 4.

    gap> QaosTransitiveGroup("d4");
    #I Retrieved 5 Transitive Groups.
    #I (C) 2004-2005 QaoS developers <kantdb@math.tu-berlin.de>,
    #I The Kant Project <kant@math.tu-berlin.de>
    #I qaos--dev--1.0--patch-32
    #I 2005-08-29 07:54:18 UTC
    <collection from database: 5 transitive groups; "d4">

Count transitive groups of degree 8.

    gap> QaosTransitiveGroup("d8",rec(Action:="count"));
    #I (C) 2004-2005 QaoS developers <kantdb@math.tu-berlin.de>,
    #I The Kant Project <kant@math.tu-berlin.de>
    #I qaos--dev--1.0--patch-33
    #I 2005-08-29 10:04:49 UTC
    50 transitive group satisfy "d8"

Retrieve the first 25 transitive groups of degree 8.

    gap> q1:=QaosTransitiveGroup("d8");
    #I Retrieved 25 Transitive Groups.
    #I (C) 2004-2005 QaoS developers <kantdb@math.tu-berlin.de>,
    #I The Kant Project <kant@math.tu-berlin.de>
    #I qaos--dev--1.0--patch-33
    #I 2005-08-29 10:08:50 UTC
    <collection from database: 25 transitive groups; "d8">

Now retrieve the next 25 results.

    gap> q2:=QaosTransitiveGroup("d8",rec(Offset:=25));
    #I Retrieved 25 Transitive Groups.
    #I (C) 2004-2005 QaoS developers <kantdb@math.tu-berlin.de>,
    #I The Kant Project <kant@math.tu-berlin.de>
    #I qaos--dev--1.0--patch-33
    #I 2005-08-29 10:09:41 UTC
    <collection from database: 25 transitive groups; "d8">

Retrieve number fields of degree 4.

    gap> QaosNumberField("d4");
    #I Retrieved 25 Algebraic Number Fields.
    #I (C) 2004-2005 QaoS developers <kantdb@math.tu-berlin.de>,
    #I The Kant Project <kant@math.tu-berlin.de>
    #I qaos--dev--1.0--patch-32
    #I 2005-08-29 07:55:11 UTC
    <collection from database: 25 number fields; "d4">

When either of these functions is called with no arguments a short general help is displayed:

    gap> QaosTransitiveGroup();
    QaosTransitiveGroup(<query> [, <optarg>]) -> <result>

    Searches the Algebraic Objects Database in Berlin.
    The query string equals the keyword search method in the web surface.
    See
    <URL>http://qaos.math.tu-berlin.de/qaos/query.scm?type=trnsg&action=Help</URL>
    for more information about the syntax and keywords.

    Note: You must have `curl' (see <URL>http://curl.haxx.se</URL>) installed and properly
    configured in order to use &QaoS; from within &GAP;.

    true

In this manual we also refer to the help screen of the web interface to obtain more detailed information about the query string.

#### 3.2 Examine the contents of a database collection

In order to actually see the information there is a function called `QaosResult` (3.2-2) which displays a list of retrieved objects.

##### 3.2-1 QaosResult

`‣ QaosResult`( database-collection ) | ( function )

Return the list of retrieved objects in database-collection. This function actually reveals the result of a query.

##### 3.2-2 QaosResult

`‣ QaosResult`( database-count ) | ( function )

Return the (non-negative) integer of a count query (see `Action`-component in optarg record).

Let us examine the groups behind a query for degree 8 abelian transitive groups.

    gap> Q:=QaosTransitiveGroup("d8 abelian");
    #I Retrieved 3 Transitive Groups.
    #I (C) 2004-2005 QaoS developers <kantdb@math.tu-berlin.de>,
    #I The Kant Project <kant@math.tu-berlin.de>
    #I qaos--dev--1.0--patch-32
    #I 2005-08-29 08:01:30 UTC
    <collection from database: 3 transitive groups; "d8 abelian">
    gap>
    gap> QaosResult(Q);
    [ <transitive group from database: C(8) = 8; id 39>,
      <transitive group from database: 4[x]2; id 40>,
      <transitive group from database: E(8) = 2[x]2[x]2; id 41> ]

To use the retrieved group objects in GAP you may use the `AsGroup` (3.2-3) operation on a result.

##### 3.2-3 AsGroup

`‣ AsGroup`( database-group-object ) | ( function )

Return the GAP group object representing database-group-object. If database-group-object cannot be turned into a group, return `fail`.

From the above example, let us GAP-ify the third group object.

    gap> AsGroup(QaosResult(Q)[3]);
    Group([ (1,8)(2,3)(4,5)(6,7), (1,3)(2,8)(4,6)(5,7), (1,5)(2,6)(3,7)(4,8) ])

For number field objects the corresponding operation is `AsField` (3.2-4).

##### 3.2-4 AsField

`‣ AsField`( database-field-object ) | ( function )

Return the GAP field object representing database-field-object. If database-field-object cannot be turned into a field, return `fail`.

Consider following example for number fields (number fields of degree 5 with absolute discrimnant between 100000 and 120000 and non-trivial class group):

    gap> R:=QaosNumberField("d5 |disc| >=100000 |disc| <=120000 cn>1");
    #I Retrieved 18 Algebraic Number Fields.
    #I (C) 2004-2005 QaoS developers <kantdb@math.tu-berlin.de>,
    #I The Kant Project <kant@math.tu-berlin.de>
    #I qaos--dev--1.0--patch-32
    #I 2005-08-29 08:32:35 UTC
    <collection from database: 18 number fields; "d5 |disc| >=100000 |disc| <=120000 cn>1">
    gap> QaosResult(R);
    [ <number field from database: x_1^5+x_1^4-2*x_1^2+7*x_1+1; id 843712>,
      <number field from database: x_1^5+x_1^4+2*x_1^3-4*x_1^2+2*x_1-1; id 843791>,
      <number field from database: x_1^5+2*x_1^4-2*x_1^3+2*x_1^2+x_1+4; id 843841>,
      <number field from database: x_1^5+x_1^4+4*x_1^3+4*x_1-1; id 843892>,
      <number field from database: x_1^5+x_1^4+x_1^3+4*x_1^2+3*x_1-1; id 843894>,
      <number field from database: x_1^5+x_1^4-3*x_1^3-2*x_1^2+6*x_1-4; id 843897>,
      <number field from database: x_1^5+2*x_1^3+2*x_1^2+4*x_1+1; id 843930>,
      <number field from database: x_1^5+2*x_1^4+6*x_1^3+4*x_1^2+4*x_1-1; id 843955>,
      <number field from database: x_1^5+x_1^4-4*x_1^2+2*x_1+1; id 844004>,
      <number field from database: x_1^5+4*x_1^2+1; id 844011>,
      <number field from database: x_1^5+2*x_1^4+3*x_1^3+2*x_1^2-3*x_1+1; id 844036>,
      <number field from database: x_1^5+x_1^4+3*x_1^3+2*x_1^2+8*x_1-1; id 844043>,
      <number field from database: x_1^5+x_1^4+5*x_1^3+5*x_1^2+2*x_1+1; id 844062>,
      <number field from database: x_1^5+2*x_1^4-3*x_1^3+6*x_1^2-3*x_1+1; id 844066>,
      <number field from database: x_1^5+x_1^4-x_1^3-4*x_1^2+5*x_1-1; id 844082>,
      <number field from database: x_1^5+2*x_1^4-x_1^3-3*x_1^2+4*x_1+1; id 844107>,
      <number field from database: x_1^5+x_1^4+2*x_1^3+3*x_1^2-2*x_1+1; id 844110>,
      <number field from database: x_1^5+2*x_1^4-6*x_1^3+5*x_1^2-2*x_1+1; id 844145> ]
    gap> AsField(QaosResult(R)[4]);
    <field in characteristic 0>

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