# Revision history [back]

Armed with some knowledge of the general structure of Sage, you can find it out:

sage: c1(3)
zeta6 - 1
sage: c1(3).parent()
Cyclotomic Field of order 6 and degree 2
sage: c(3).parent().gens()
(zeta6,)
sage: c(3).minpoly()
x^2 - x + 1
sage: cyclotomic_polynomial(6)
x^2 - x + 1


Armed with some knowledge of the general structure of Sage, you can find it out:

sage: c1(3)
zeta6 - 1
sage: c1(3).parent()
Cyclotomic Field of order 6 and degree 2
sage: c(3).parent().gens()
(zeta6,)
sage: c(3).parent().gens()[0].minpoly()
x^2 - x + 1
sage: c(3).minpoly()
x^2 - x + 1
sage: cyclotomic_polynomial(6)
x^2 - x + 1


Armed with some knowledge of the general structure of Sage, you can find it out:

sage: c1(3)
zeta6 - 1
sage: c1(3).parent()
Cyclotomic Field of order 6 and degree 2
sage: c(3).parent().gens()
(zeta6,)
sage: c(3).parent().gens()[0].minpoly()
x^2 - x + 1
sage: c(3).minpoly()
cyclotomic_polynomial(6)
x^2 - x + 1
sage: cyclotomic_polynomial(6)
c(3).minpoly()
x^2 - + x + 1


Armed with some knowledge of the general structure of Sage, you can find it out:

sage: c1(3)
zeta6 - 1
sage: c1(3).parent()
Cyclotomic Field of order 6 and degree 2
sage: c(3).parent().gens()
c1(3).parent().gens()
(zeta6,)
sage: c(3).parent().gens()[0].minpoly()
c1(3).parent().gens()[0].minpoly()
x^2 - x + 1
sage: cyclotomic_polynomial(6)
x^2 - x + 1
sage: c(3).minpoly()
c1(3).minpoly()
x^2 + x + 1
sage: c1(3).is_unit()
True


Armed with some knowledge of the general structure of Sage, you can find it out:

sage: c1(3)
zeta6 - 1
sage: c1(3).parent()
K = c1(3).parent(); K
Cyclotomic Field of order 6 and degree 2
sage: c1(3).parent().gens()
K.gens()
(zeta6,)
sage: c1(3).parent().gens()[0].minpoly()
zeta6 = K.gens()[0]
sage: zeta6.minpoly()
x^2 - x + 1
sage: cyclotomic_polynomial(6)
x^2 - x + 1
sage: c1(3).minpoly()
x^2 + x + 1
sage: c1(3).is_unit()
True
UK = K.unit_group()
Unit group with structure C6 of Cyclotomic Field of order 6 and degree 2
sage: UK.log(c1(3))
(2,)
sage: zeta6^2
zeta6 - 1


Armed with some knowledge of the general structure of Sage, you can find it out:

sage: c1(3)
zeta6 - 1
sage: K = c1(3).parent(); K
Cyclotomic Field of order 6 and degree 2
sage: K.gens()
(zeta6,)
sage: zeta6 = K.gens()[0]
sage: zeta6.minpoly()
x^2 - x + 1
sage: cyclotomic_polynomial(6)
x^2 - x + 1
sage: c1(3).minpoly()
x^2 + x + 1
sage: UK = K.unit_group()
Unit group with structure C6 of Cyclotomic Field of order 6 and degree 2
sage: UK.gens_values()
[zeta6]
sage: c1(3) in UK
True
sage: UK.log(c1(3))
(2,)
sage: zeta6^2
zeta6 - 1