Global number field 40.0.791717805254439023624865699561776475898803884688668051353443161.1: \(\Q(\zeta_{41})\)

• Label: 40.0.79171780525...3161.1
• Degree: $40$
• Signature: $[0, 20]$
• Discriminant: $41^{39}$
• Root discriminant: $37.36$
• Ramified prime: $41$
• Class number: $121$ (GRH)
• Class group: $[11, 11]$ (GRH)
• Galois group: $C_{40}$ (as 40T1)

Normalized defining polynomial

\( x^{40} - x^{39} + x^{38} - x^{37} + x^{36} - x^{35} + x^{34} - x^{33} + x^{32} - x^{31} + x^{30} - x^{29} + x^{28} - x^{27} + x^{26} - x^{25} + x^{24} - x^{23} + x^{22} - x^{21} + x^{20} - x^{19} + x^{18} - x^{17} + x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \)

Invariants

Degree:		$40$
Signature:		$[0, 20]$
Discriminant:		\(791717805254439023624865699561776475898803884688668051353443161=41^{39}\)
Root discriminant:		$37.36$
Ramified primes:		$41$
This field is Galois and abelian over $\Q$.
Conductor:		\(41\)
Dirichlet character group:		$\lbrace$$\chi_{41}(1,·)$, $\chi_{41}(2,·)$, $\chi_{41}(3,·)$, $\chi_{41}(4,·)$, $\chi_{41}(5,·)$, $\chi_{41}(6,·)$, $\chi_{41}(7,·)$, $\chi_{41}(8,·)$, $\chi_{41}(9,·)$, $\chi_{41}(10,·)$, $\chi_{41}(11,·)$, $\chi_{41}(12,·)$, $\chi_{41}(13,·)$, $\chi_{41}(14,·)$, $\chi_{41}(15,·)$, $\chi_{41}(16,·)$, $\chi_{41}(17,·)$, $\chi_{41}(18,·)$, $\chi_{41}(19,·)$, $\chi_{41}(20,·)$, $\chi_{41}(21,·)$, $\chi_{41}(22,·)$, $\chi_{41}(23,·)$, $\chi_{41}(24,·)$, $\chi_{41}(25,·)$, $\chi_{41}(26,·)$, $\chi_{41}(27,·)$, $\chi_{41}(28,·)$, $\chi_{41}(29,·)$, $\chi_{41}(30,·)$, $\chi_{41}(31,·)$, $\chi_{41}(32,·)$, $\chi_{41}(33,·)$, $\chi_{41}(34,·)$, $\chi_{41}(35,·)$, $\chi_{41}(36,·)$, $\chi_{41}(37,·)$, $\chi_{41}(38,·)$, $\chi_{41}(39,·)$, $\chi_{41}(40,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$

Class group and class number

$C_{11}\times C_{11}$, which has order $121$ (assuming GRH)

Unit group

Rank:		$19$
Torsion generator:		\( a \) (order $82$)
Fundamental units:		Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
Regulator:		\( 310417721980536.1 \) (assuming GRH)

Galois group

$C_{40}$ (as 40T1):

A cyclic group of order 40

The 40 conjugacy class representatives for $C_{40}$

Character table for $C_{40}$ is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 5.5.2825761.1, 8.0.194754273881.1, 10.10.327381934393961.1, \(\Q(\zeta_{41})^+\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	$20^{2}$	${\href{/LocalNumberField/3.8.0.1}{8} }^{5}$	$20^{2}$	$40$	$40$	$40$	$40$	$40$	${\href{/LocalNumberField/23.10.0.1}{10} }^{4}$	$40$	${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$	${\href{/LocalNumberField/37.5.0.1}{5} }^{8}$	R	$20^{2}$	$40$	$40$	${\href{/LocalNumberField/59.5.0.1}{5} }^{8}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
41	Data not computed