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
>>> from sage.all import *
>>> J = JonesDatabase() # optional - database
>>> J # optional - database
John Jones's table of number fields with bounded ramification and degree <= 6
J = JonesDatabase() # optional - database J # optional - database
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)]
>>> from sage.all import *
>>> [(k.degree(), k.disc()) for k in J.unramified_outside([Integer(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)]
[(k.degree(), k.disc()) for k in J.unramified_outside([2])] # optional - database
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]
>>> from sage.all import *
>>> [k.disc() for k in J.unramified_outside([Integer(2)],Integer(2))] # optional - database
[8, -4, -8]
[k.disc() for k in J.unramified_outside([2],2)] # optional - database
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
>>> from sage.all import *
>>> [k.disc() for k in J.ramified_at([Integer(3),Integer(5)],Integer(3))] # optional - database
[-6075, -6075, -675, -135]
>>> factor(Integer(6075))
3^5 * 5^2
>>> factor(Integer(675))
3^3 * 5^2
>>> factor(Integer(135))
3^3 * 5
[k.disc() for k in J.ramified_at([3,5],3)] # optional - database factor(6075) factor(675) factor(135)
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]
>>> from sage.all import *
>>> J.ramified_at(Integer(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]
J.ramified_at(101) # optional - database
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
>>> from sage.all import *
>>> K = QuadraticField(Integer(29), 'x')
>>> K.class_number()
1
>>> K = QuadraticField(Integer(65), 'x')
>>> K.class_number()
2
>>> K = QuadraticField(-Integer(11), 'x')
>>> K.class_number()
1
>>> K = QuadraticField(-Integer(15), 'x')
>>> K.class_number()
2
>>> 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
>>> K = QuadraticField(Integer(401), 'x')
>>> 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?
>>> K.class_number()
5
>>> K.discriminant()
401
>>> K = QuadraticField(-Integer(479), 'x')
>>> 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
>>> K.class_number()
25
>>> K.pari_polynomial()
x^2 + 479
>>> K.degree()
2
K = QuadraticField(29, 'x') K.class_number() K = QuadraticField(65, 'x') K.class_number() K = QuadraticField(-11, 'x') K.class_number() K = QuadraticField(-15, 'x') K.class_number() K.class_group() K = QuadraticField(401, 'x') K.class_group() K.class_number() K.discriminant() K = QuadraticField(-479, 'x') K.class_group() K.class_number() K.pari_polynomial() K.degree()
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
>>> from sage.all import *
>>> x = PolynomialRing(QQ, 'x').gen()
>>> K = NumberField(x**Integer(5)+Integer(10)*x+Integer(1), 'a')
>>> K
Number Field in a with defining polynomial x^5 + 10*x + 1
>>> K.degree()
5
>>> K.pari_polynomial()
x^5 + 10*x + 1
>>> K.discriminant()
25603125
>>> K.class_group()
Class group of order 1 of Number Field in a with defining
polynomial x^5 + 10*x + 1
>>> K.class_number()
1
x = PolynomialRing(QQ, 'x').gen() K = NumberField(x^5+10*x+1, 'a') K K.degree() K.pari_polynomial() K.discriminant() K.class_group() K.class_number()
See also the link for class numbers at http://mathworld.wolfram.com/ClassNumber.html at the Math World site for tables, formulas, and background information.
For cyclotomic fields, try:
sage: K = CyclotomicField(19) sage: K.class_number() # long time 1
>>> from sage.all import * >>> K = CyclotomicField(Integer(19)) >>> K.class_number() # long time 1
K = CyclotomicField(19) K.class_number() # long time
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]
>>> from sage.all import *
>>> x = PolynomialRing(QQ, 'x').gen()
>>> K = NumberField(x**Integer(5)+Integer(10)*x+Integer(1), 'a')
>>> K.integral_basis()
[1, a, a^2, a^3, a^4]
x = PolynomialRing(QQ, 'x').gen() K = NumberField(x^5+10*x+1, 'a') K.integral_basis()
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]
>>> from sage.all import *
>>> x = PolynomialRing(QQ, 'x').gen()
>>> K = NumberField(x**Integer(3) + x**Integer(2) - Integer(2)*x + Integer(8), 'a')
>>> K.integral_basis()
[1, 1/2*a^2 + 1/2*a, a^2]
x = PolynomialRing(QQ, 'x').gen() K = NumberField(x^3 + x^2 - 2*x + 8, 'a') K.integral_basis()