Class groups

This module allows to work with ideal classes and compute the class numbers of quadratic integer rings.

To compute operations with ideal classes within the same quadratic ring, all ideal classes must be created as IdealClass objects. To do so:

1. Create the quadratic integer ring \(\mathcal{O}_{\mathbb{Q}[\sqrt{d}]}\) with the function QuadraticIntegerRing.

   >>> O = QuadraticIntegerRing(-5)
   

2. Create the representative ideal of the ideal class.

   >>> generator1 = O(3)
   >>> generator2 = O("1 + sqrt(-5)")
   >>> I = Ideal(generator1, generator2)
   

3. Create the ideal class object with the ideal as argument and use the available operators and methods.

   >>> a = IdealClass(I)
   >>> a.order
   2
   

To compute the class group of a quadratic integer ring (i.e. the class number and the generators of the class group), simply instance the class ClassGroup and access its attributes class_number and generators.

Note that this module, class_group, need to be imported to use its classes and functions. There are several ways to import it:

1. Import all functions and classes of QiPy:

   >>> from qipy import *
   >>> O = QuadraticIntegerRing(-5)
   >>> a = IdealClass( Ideal(O(2), O("1 + sqrt(-5)")) )
   

2. Import only the package, and use the package’s identifier to access the classes and functions:

   >>> import qipy
   >>> O = qipy.QuadraticIntegerRing(-5)
   >>> I = qipy.Ideal(O(2), O("1 + sqrt(-5)"))
   >>> a = qipy.IdealClass(I)
   

---

class IdealClass(ideal, name=None)

Represent an ideal class of a quadratic integer ring.

>>> Zi = QuadraticIntegerRing(-1)
>>> I = Ideal(Zi(3))
>>> a = IdealClass(I)
>>> a
[<3>]
>>> a == a * a
True
>>> O = QuadraticIntegerRing(-5)
>>> J = Ideal(O(2), O("1 + sqrt(-5)"))
>>> b = IdealClass(J)
>>> b
[<1 + sqrt(5)*I,2*sqrt(5)*I>]
>>> b ** 2
[<-2,2*sqrt(5)*I>]

This class supports the operators * and ** with their natural meaning.

Parameters:

• ideal – a representative of the ideal class
• name – an optional name for the representative.

O

the quadratic integer ring related to the ideal class.

representative

a representative of the ideal class.

inverse()

Return the inverse of the ideal class.

>>> Zi = QuadraticIntegerRing(-1)
>>> a = IdealClass(Ideal(Zi(3)))
>>> a
[<3>]
>>> a.inverse()
[<1>]
>>> O = QuadraticIntegerRing(-5)
>>> b = IdealClass(Ideal(O(2), O("1 + sqrt(-5)")))
>>> b
[<1 + sqrt(5)*I,2*sqrt(5)*I>]
>>> b.inverse()
[<1 + sqrt(5)*I,2*sqrt(5)*I>]

is_trivial()

Test whether the ideal class is the trivial class.

>>> Zi = QuadraticIntegerRing(-1)
>>> a = IdealClass(Ideal(Zi(3)))
>>> a
[<3>]
>>> a.is_trivial()
True
>>> O = QuadraticIntegerRing(-5)
>>> b = IdealClass(Ideal(O(2), O("1 + sqrt(-5)")))
>>> b
[<1 + sqrt(5)*I,2*sqrt(5)*I>]
>>> b.is_trivial()
False

order

The order of the ideal class, that is, the smallest positive intenger such that self ** order is the trivial class.

>>> Zi = QuadraticIntegerRing(-1)
>>> a = IdealClass(Ideal(Zi(3)))
>>> a
[<3>]
>>> a.order
1
>>> O = QuadraticIntegerRing(-5)
>>> b = IdealClass(Ideal(O(2), O("1 + sqrt(-5)")))
>>> b
[<1 + sqrt(5)*I,2*sqrt(5)*I>]
>>> b.order
2

classmethod trivial_class(d)

Return the trivial class (with the unit ideal as representative) of the quadratic integer ring defined by \(d\).

>>> IdealClass.trivial_class(-1)
[<1>]

Returns:the trivial ideal class. Return type:IdealClass

class ClassGroup(d, verbose=False)

Represent the class group of a quadratic integer ring.

>>> G = ClassGroup(-1)
>>> G.class_number
1
>>> G.generators
[[<1>]]
>>> H = ClassGroup(-14, verbose=True)  
List of generators (including dependent): [[p2], [p3]]
Testing whether [p3] is dependent:
    [p3] is independent
Testing whether [p2] is dependent:
    Dependency relation found: [<1>] == [p2] * [p3]^2
    [p2] removed from the list of generators
Class number: 4
Generators: [[p3]]

The non-trivial generators are represented as [pX] where pX is a prime ideal that divides \(\langle X \rangle\) (the ideal generated by \(X\)).

Warning

In the of the object, the generators and the class number are computed. Therefore, the creation of the object may take some time.

Parameters:

• d – a non-square free integer that defines the quadratic integer ring.
• verbose – if True, information about the computation process of the class group is printed.

O

the quadratic integer ring related to the ideal class.

generators

a list of ideal classes that span the class group.

class_number

the order of the class group.