Global number field 40.40.870502932794550416715512205314557822885758691905429612124950249853404839936.1: \(\Q(\zeta_{164})^+\)

• Label: 40.40.8705029327...9936.1
• Degree: $40$
• Signature: $[40, 0]$
• Discriminant: $2^{40}\cdot 41^{39}$
• Root discriminant: $74.73$
• Ramified primes: $2, 41$
• Class number: $1$ (GRH)
• Class group: Trivial (GRH)
• Galois group: $C_{40}$ (as 40T1)

Normalized defining polynomial

\( x^{40} - 41 x^{38} + 779 x^{36} - 9102 x^{34} + 73185 x^{32} - 429352 x^{30} + 1901416 x^{28} - 6487184 x^{26} + 17250012 x^{24} - 35937525 x^{22} + 58659315 x^{20} - 74657310 x^{18} + 73370115 x^{16} - 54826020 x^{14} + 30458900 x^{12} - 12183560 x^{10} + 3350479 x^{8} - 591261 x^{6} + 59983 x^{4} - 2870 x^{2} + 41 \)

Invariants

Degree:		$40$
Signature:		$[40, 0]$
Discriminant:		\(870502932794550416715512205314557822885758691905429612124950249853404839936=2^{40}\cdot 41^{39}\)
Root discriminant:		$74.73$
Ramified primes:		$2, 41$
This field is Galois and abelian over $\Q$.
Conductor:		\(164=2^{2}\cdot 41\)
Dirichlet character group:		$\lbrace$$\chi_{164}(1,·)$, $\chi_{164}(3,·)$, $\chi_{164}(5,·)$, $\chi_{164}(7,·)$, $\chi_{164}(9,·)$, $\chi_{164}(11,·)$, $\chi_{164}(141,·)$, $\chi_{164}(15,·)$, $\chi_{164}(19,·)$, $\chi_{164}(21,·)$, $\chi_{164}(151,·)$, $\chi_{164}(25,·)$, $\chi_{164}(27,·)$, $\chi_{164}(133,·)$, $\chi_{164}(33,·)$, $\chi_{164}(35,·)$, $\chi_{164}(37,·)$, $\chi_{164}(135,·)$, $\chi_{164}(45,·)$, $\chi_{164}(47,·)$, $\chi_{164}(49,·)$, $\chi_{164}(55,·)$, $\chi_{164}(57,·)$, $\chi_{164}(61,·)$, $\chi_{164}(63,·)$, $\chi_{164}(67,·)$, $\chi_{164}(71,·)$, $\chi_{164}(73,·)$, $\chi_{164}(75,·)$, $\chi_{164}(77,·)$, $\chi_{164}(79,·)$, $\chi_{164}(81,·)$, $\chi_{164}(95,·)$, $\chi_{164}(99,·)$, $\chi_{164}(105,·)$, $\chi_{164}(111,·)$, $\chi_{164}(113,·)$, $\chi_{164}(147,·)$, $\chi_{164}(121,·)$, $\chi_{164}(125,·)$$\rbrace$
This is not 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

Trivial group, which has order $1$ (assuming GRH)

Unit group

Rank:		$39$
Torsion generator:		\( -1 \) (order $2$)
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:		\( 6180661033584260000000000 \) (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, 8.8.49857094113536.1, 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	R	${\href{/LocalNumberField/3.8.0.1}{8} }^{5}$	$20^{2}$	$40$	$40$	$40$	$40$	$40$	${\href{/LocalNumberField/23.5.0.1}{5} }^{8}$	$40$	${\href{/LocalNumberField/31.5.0.1}{5} }^{8}$	${\href{/LocalNumberField/37.5.0.1}{5} }^{8}$	R	$20^{2}$	$40$	$40$	${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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
2	Data not computed
41	Data not computed