Global number field 46.46.165269405800315006446654642733283089940652236420810887453559572571427563258244528313989.1: \(\Q(\zeta_{141})^+\)

• Label: 46.46.1652694058...3989.1
• Degree: $46$
• Signature: $[46, 0]$
• Discriminant: $3^{23}\cdot 47^{45}$
• Root discriminant: $74.87$
• Ramified primes: $3, 47$
• Class number: $1$ (GRH)
• Class group: Trivial (GRH)
• Galois group: $C_{46}$ (as 46T1)

Normalized defining polynomial

\( x^{46} - x^{45} - 46 x^{44} + 46 x^{43} + 988 x^{42} - 988 x^{41} - 13159 x^{40} + 13159 x^{39} + 121731 x^{38} - 121731 x^{37} - 830207 x^{36} + 830207 x^{35} + 4324189 x^{34} - 4324189 x^{33} - 17581994 x^{32} + 17581994 x^{31} + 56562010 x^{30} - 56562010 x^{29} - 145057650 x^{28} + 145057650 x^{27} + 297415766 x^{26} - 297415766 x^{25} - 486968926 x^{24} + 486968926 x^{23} + 633580634 x^{22} - 633580634 x^{21} - 649220446 x^{20} + 649220446 x^{19} + 516962354 x^{18} - 516962354 x^{17} - 313942891 x^{16} + 313942891 x^{15} + 141714824 x^{14} - 141714824 x^{13} - 45908941 x^{12} + 45908941 x^{11} + 10162529 x^{10} - 10162529 x^{9} - 1431196 x^{8} + 1431196 x^{7} + 114634 x^{6} - 114634 x^{5} - 4276 x^{4} + 4276 x^{3} + 48 x^{2} - 48 x + 1 \)

Invariants

Degree:		$46$
Signature:		$[46, 0]$
Discriminant:		\(165269405800315006446654642733283089940652236420810887453559572571427563258244528313989=3^{23}\cdot 47^{45}\)
Root discriminant:		$74.87$
Ramified primes:		$3, 47$
This field is Galois and abelian over $\Q$.
Conductor:		\(141=3\cdot 47\)
Dirichlet character group:		$\lbrace$$\chi_{141}(1,·)$, $\chi_{141}(130,·)$, $\chi_{141}(4,·)$, $\chi_{141}(5,·)$, $\chi_{141}(134,·)$, $\chi_{141}(7,·)$, $\chi_{141}(136,·)$, $\chi_{141}(137,·)$, $\chi_{141}(11,·)$, $\chi_{141}(140,·)$, $\chi_{141}(16,·)$, $\chi_{141}(20,·)$, $\chi_{141}(23,·)$, $\chi_{141}(25,·)$, $\chi_{141}(26,·)$, $\chi_{141}(28,·)$, $\chi_{141}(29,·)$, $\chi_{141}(34,·)$, $\chi_{141}(35,·)$, $\chi_{141}(37,·)$, $\chi_{141}(38,·)$, $\chi_{141}(41,·)$, $\chi_{141}(44,·)$, $\chi_{141}(49,·)$, $\chi_{141}(55,·)$, $\chi_{141}(61,·)$, $\chi_{141}(62,·)$, $\chi_{141}(64,·)$, $\chi_{141}(77,·)$, $\chi_{141}(79,·)$, $\chi_{141}(80,·)$, $\chi_{141}(86,·)$, $\chi_{141}(92,·)$, $\chi_{141}(97,·)$, $\chi_{141}(100,·)$, $\chi_{141}(103,·)$, $\chi_{141}(104,·)$, $\chi_{141}(106,·)$, $\chi_{141}(107,·)$, $\chi_{141}(112,·)$, $\chi_{141}(113,·)$, $\chi_{141}(115,·)$, $\chi_{141}(116,·)$, $\chi_{141}(118,·)$, $\chi_{141}(121,·)$, $\chi_{141}(125,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $a^{42}$, $a^{43}$, $a^{44}$, $a^{45}$

Class group and class number

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

Unit group

Rank:		$45$
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:		\( 45504514750422600000000000000 \) (assuming GRH)

Galois group

$C_{46}$ (as 46T1):

A cyclic group of order 46

The 46 conjugacy class representatives for $C_{46}$

Character table for $C_{46}$ is not computed

Intermediate fields

\(\Q(\sqrt{141}) \), \(\Q(\zeta_{47})^+\)

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	$46$	R	$23^{2}$	$23^{2}$	$23^{2}$	$46$	$46$	$46$	$23^{2}$	$23^{2}$	$46$	$23^{2}$	$23^{2}$	$46$	R	$46$	$46$

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	Data not computed
47	Data not computed