Understanding Output in SageMath Regarding Dirichlet Characters

DirichletGroup

asked 2018-12-05 18:17:59 +0100

Nicklovn
13 ●3

updated 2021-07-15 08:15:03 +0100

FrédéricC

5206 ●3 ●43 ●114

 p=7
G = DirichletGroup(p); G

m=3; n=ZZ((p-1)/m); print m,n

c=G[1]

c1=c^n;c1

The output is:

Dirichlet character modulo 7 of conductor 7 mapping 3 |--> zeta6 - 1

Can anyone explain what zeta6 is? Is this the Riemann-Zeta function? Is this the whole group of units? Is there a relation to the Eisenstein primes? I'm still a bit weak in this material and am having trouble grasping some of these sage outputs. Thank you in advance!

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kcrisman ( 2018-12-07 19:14:46 +0100 )edit

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answered 2018-12-05 19:42:54 +0100

rburing

10909 ●4 ●79 ●218 https://www.rburing.nl/

updated 2018-12-05 20:01:37 +0100

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

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Asked: 2018-12-05 18:17:59 +0100

Seen: 289 times

Last updated: Dec 05 '18

Understanding Output in SageMath Regarding Dirichlet Characters edit