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sage: 
sage: K1=QQ[(-2003)^(1/2)]
sage: K1
Number Field in a with defining polynomial x^2 + 2003
sage: K1.class_group()
Class group of order 9 with structure C9 of Number Field in a with defining polynomial x^2 + 2003
sage: K2=QQ[(-3*5*7)^(1/2)].class_group()
sage: K2
Class group of order 8 with structure C2 x C2 x C2 of Number Field in a with defining polynomial x^2 + 105
sage: K2=QQ[(-3*5*7)^(1/2)]
sage: K = QQ[sqrt(-21)]
sage: K.class_group()
Class group of order 4 with structure C2 x C2 of Number Field in a with defining polynomial x^2 + 21
sage: G=K.class_group()
sage: G.gens()
(Fractional ideal class (5, a + 2), Fractional ideal class (2, a + 1))
sage: q2 = K.factor(2)[0][0]
sage: q2
Fractional ideal (2, a + 1)
sage: q3 = K.factor(3)[0][0]
sage: q3
Fractional ideal (3, a)
sage: q5 = K.factor(5)[0][0]
sage: q5
Fractional ideal (5, a + 2)
sage: G(q2)
Fractional ideal class (2, a + 1)
sage: q2^2
Fractional ideal (2)
sage: G(q2)^2
Trivial principal fractional ideal class
sage: a = G(q2)
sage: b = G(q3)
sage: c = G(q5)
sage: a*b*c
Trivial principal fractional ideal class
sage: K1
Number Field in a with defining polynomial x^2 + 2003
sage: K1.class_group()
Class group of order 9 with structure C9 of Number Field in a with defining polynomial x^2 + 2003
sage: K1.class_group().gens()
(Fractional ideal class (3, 1/2*a - 1/2),)
sage: K2
Number Field in a with defining polynomial x^2 + 105
sage: K2.class_group()
Class group of order 8 with structure C2 x C2 x C2 of Number Field in a with defining polynomial x^2 + 105
sage: K2.class_group().gens()
(Fractional ideal class (11, a + 7),
 Fractional ideal class (10, a + 5),
 Fractional ideal class (6, a + 3))
sage: K.minkowski_bound()
4*sqrt(21)/pi
sage: n(K.minkowski_bound())
5.83471659156000
sage: n(K1.minkowski_bound())
28.4918466116609
sage: n(K2.minkowski_bound())
13.0468229281740
sage: K1
Number Field in a with defining polynomial x^2 + 2003
sage: