Global number field 40.40.2760549293355133823580852196951954752012701839690650502926252551414931561.1: \(\Q(\zeta_{123})^+\)

• Label: 40.40.2760549293...1561.1
• Degree: $40$
• Signature: $[40, 0]$
• Discriminant: $3^{20}\cdot 41^{39}$
• Root discriminant: $64.72$
• Ramified primes: $3, 41$
• Class number: $1$ (GRH)
• Class group: Trivial (GRH)
• Galois group: $C_{40}$ (as 40T1)

Normalized defining polynomial

\( x^{40} - x^{39} - 40 x^{38} + 40 x^{37} + 739 x^{36} - 739 x^{35} - 8363 x^{34} + 8363 x^{33} + 64822 x^{32} - 64822 x^{31} - 364530 x^{30} + 364530 x^{29} + 1536886 x^{28} - 1536886 x^{27} - 4950298 x^{26} + 4950298 x^{25} + 12299714 x^{24} - 12299714 x^{23} - 23637811 x^{22} + 23637811 x^{21} + 35021504 x^{20} - 35021504 x^{19} - 39635806 x^{18} + 39635806 x^{17} + 33734309 x^{16} - 33734309 x^{15} - 21091711 x^{14} + 21091711 x^{13} + 9367189 x^{12} - 9367189 x^{11} - 2816371 x^{10} + 2816371 x^{9} + 534108 x^{8} - 534108 x^{7} - 57153 x^{6} + 57153 x^{5} + 2830 x^{4} - 2830 x^{3} - 40 x^{2} + 40 x + 1 \)

Invariants

Degree:		$40$
Signature:		$[40, 0]$
Discriminant:		\(2760549293355133823580852196951954752012701839690650502926252551414931561=3^{20}\cdot 41^{39}\)
Root discriminant:		$64.72$
Ramified primes:		$3, 41$
This field is Galois and abelian over $\Q$.
Conductor:		\(123=3\cdot 41\)
Dirichlet character group:		$\lbrace$$\chi_{123}(1,·)$, $\chi_{123}(4,·)$, $\chi_{123}(10,·)$, $\chi_{123}(11,·)$, $\chi_{123}(14,·)$, $\chi_{123}(16,·)$, $\chi_{123}(17,·)$, $\chi_{123}(25,·)$, $\chi_{123}(26,·)$, $\chi_{123}(29,·)$, $\chi_{123}(31,·)$, $\chi_{123}(35,·)$, $\chi_{123}(37,·)$, $\chi_{123}(38,·)$, $\chi_{123}(40,·)$, $\chi_{123}(43,·)$, $\chi_{123}(44,·)$, $\chi_{123}(46,·)$, $\chi_{123}(47,·)$, $\chi_{123}(49,·)$, $\chi_{123}(53,·)$, $\chi_{123}(56,·)$, $\chi_{123}(61,·)$, $\chi_{123}(64,·)$, $\chi_{123}(65,·)$, $\chi_{123}(68,·)$, $\chi_{123}(71,·)$, $\chi_{123}(73,·)$, $\chi_{123}(89,·)$, $\chi_{123}(91,·)$, $\chi_{123}(95,·)$, $\chi_{123}(100,·)$, $\chi_{123}(101,·)$, $\chi_{123}(103,·)$, $\chi_{123}(104,·)$, $\chi_{123}(110,·)$, $\chi_{123}(115,·)$, $\chi_{123}(116,·)$, $\chi_{123}(118,·)$, $\chi_{123}(121,·)$$\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:		\( 317481165057225300000000 \) (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, 8.8.15775096184361.1, 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	$20^{2}$	R	$20^{2}$	$40$	$40$	$40$	$40$	$40$	${\href{/LocalNumberField/23.5.0.1}{5} }^{8}$	$40$	${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$	${\href{/LocalNumberField/37.5.0.1}{5} }^{8}$	R	$20^{2}$	$40$	$40$	${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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
$3$	3.8.4.2	$x^{8} - 27 x^{2} + 162$	$2$	$4$	$4$	$C_8$	$[\ ]_{2}^{4}$
	3.8.4.2	$x^{8} - 27 x^{2} + 162$	$2$	$4$	$4$	$C_8$	$[\ ]_{2}^{4}$
	3.8.4.2	$x^{8} - 27 x^{2} + 162$	$2$	$4$	$4$	$C_8$	$[\ ]_{2}^{4}$
	3.8.4.2	$x^{8} - 27 x^{2} + 162$	$2$	$4$	$4$	$C_8$	$[\ ]_{2}^{4}$
	3.8.4.2	$x^{8} - 27 x^{2} + 162$	$2$	$4$	$4$	$C_8$	$[\ ]_{2}^{4}$
41	Data not computed