Number fields

Ramification

How do you compute the number fields with given discriminant and ramification in Sage?

Sage can access the Jones database of number fields with bounded ramification and degree less than or equal to 6. It must be installed separately (database_jones_numfield).

First load the database:

sage: J = JonesDatabase()            # optional - database
sage: J                              # optional - database
John Jones's table of number fields with bounded ramification and degree <= 6

List the degree and discriminant of all fields in the database that have ramification at most at 2:

sage: [(k.degree(), k.disc()) for k in J.unramified_outside([2])] # optional - database
[(4, -2048), (2, 8), (4, -1024), (1, 1), (4, 256), (2, -4), (4, 2048), (4, 512), (4, 2048), (2, -8), (4, 2048)]

List the discriminants of the fields of degree exactly 2 unramified outside 2:

sage: [k.disc() for k in J.unramified_outside([2],2)] # optional - database
[8, -4, -8]

List the discriminants of cubic field in the database ramified exactly at 3 and 5:

sage: [k.disc() for k in J.ramified_at([3,5],3)] # optional - database
[-6075, -6075, -675, -135]
sage: factor(6075)
3^5 * 5^2
sage: factor(675)
3^3 * 5^2
sage: factor(135)
3^3 * 5

List all fields in the database ramified at 101:

sage: J.ramified_at(101)                     # optional - database
[Number Field in a with defining polynomial x^2 - 101,
 Number Field in a with defining polynomial x^4 - x^3 + 13*x^2 - 19*x + 361,
 Number Field in a with defining polynomial x^5 - x^4 - 40*x^3 - 93*x^2 - 21*x + 17,
 Number Field in a with defining polynomial x^5 + x^4 - 6*x^3 - x^2 + 18*x + 4,
 Number Field in a with defining polynomial x^5 + 2*x^4 + 7*x^3 + 4*x^2 + 11*x - 6]

Class numbers

How do you compute the class number of a number field in Sage?

The class_number is a method associated to a QuadraticField object:

sage: K = QuadraticField(29, 'x')
sage: K.class_number()
1
sage: K = QuadraticField(65, 'x')
sage: K.class_number()
2
sage: K = QuadraticField(-11, 'x')
sage: K.class_number()
1
sage: K = QuadraticField(-15, 'x')
sage: K.class_number()
2
sage: K.class_group()
Class group of order 2 with structure C2 of Number Field in x with defining polynomial x^2 + 15 with x = 3.872983346207417?*I
sage: K = QuadraticField(401, 'x')
sage: K.class_group()
Class group of order 5 with structure C5 of Number Field in x with defining polynomial x^2 - 401 with x = 20.02498439450079?
sage: K.class_number()
5
sage: K.discriminant()
401
sage: K = QuadraticField(-479, 'x')
sage: K.class_group()
Class group of order 25 with structure C25 of Number Field in x with defining polynomial x^2 + 479 with x = 21.88606862823929?*I
sage: K.class_number()
25
sage: K.pari_polynomial()
x^2 + 479
sage: K.degree()
2

Here’s an example involving a more general type of number field:

sage: x = PolynomialRing(QQ, 'x').gen()
sage: K = NumberField(x^5+10*x+1, 'a')
sage: K
Number Field in a with defining polynomial x^5 + 10*x + 1
sage: K.degree()
5
sage: K.pari_polynomial()
x^5 + 10*x + 1
sage: K.discriminant()
25603125
sage: K.class_group()
Class group of order 1 of Number Field in a with defining
polynomial x^5 + 10*x + 1
sage: K.class_number()
1

  • For cyclotomic fields, try:

    sage: K = CyclotomicField(19)
sage: K.class_number()    # long time
1

For further details, see the documentation strings in the ring/number_field.py file.

Integral basis

How do you compute an integral basis of a number field in Sage?

Sage can compute a list of elements of this number field that are a basis for the full ring of integers of a number field.

sage: x = PolynomialRing(QQ, 'x').gen()
sage: K = NumberField(x^5+10*x+1, 'a')
sage: K.integral_basis()
[1, a, a^2, a^3, a^4]

Next we compute the ring of integers of a cubic field in which 2 is an “essential discriminant divisor”, so the ring of integers is not generated by a single element.

sage: x = PolynomialRing(QQ, 'x').gen()
sage: K = NumberField(x^3 + x^2 - 2*x + 8, 'a')
sage: K.integral_basis()
[1, 1/2*a^2 + 1/2*a, a^2]