Global number field 46.0.1755511210260049172778020908173078657717675374080672665297567056535308458607.1: \(\Q(\zeta_{47})\)

• Label: 46.0.17555112102...8607.1
• Degree: $46$
• Signature: $[0, 23]$
• Discriminant: $-\,47^{45}$
• Root discriminant: $43.23$
• Ramified prime: $47$
• Class number: $695$ (GRH)
• Class group: $[695]$ (GRH)
• Galois group: $C_{46}$ (as 46T1)

Normalized defining polynomial

\( x^{46} - x^{45} + x^{44} - x^{43} + x^{42} - x^{41} + x^{40} - x^{39} + x^{38} - x^{37} + x^{36} - x^{35} + x^{34} - x^{33} + x^{32} - x^{31} + x^{30} - x^{29} + x^{28} - x^{27} + x^{26} - x^{25} + x^{24} - x^{23} + x^{22} - x^{21} + x^{20} - x^{19} + x^{18} - x^{17} + x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \)

Invariants

Degree:		$46$
Signature:		$[0, 23]$
Discriminant:		\(-1755511210260049172778020908173078657717675374080672665297567056535308458607=-\,47^{45}\)
Root discriminant:		$43.23$
Ramified primes:		$47$
This field is Galois and abelian over $\Q$.
Conductor:		\(47\)
Dirichlet character group:		$\lbrace$$\chi_{47}(1,·)$, $\chi_{47}(2,·)$, $\chi_{47}(3,·)$, $\chi_{47}(4,·)$, $\chi_{47}(5,·)$, $\chi_{47}(6,·)$, $\chi_{47}(7,·)$, $\chi_{47}(8,·)$, $\chi_{47}(9,·)$, $\chi_{47}(10,·)$, $\chi_{47}(11,·)$, $\chi_{47}(12,·)$, $\chi_{47}(13,·)$, $\chi_{47}(14,·)$, $\chi_{47}(15,·)$, $\chi_{47}(16,·)$, $\chi_{47}(17,·)$, $\chi_{47}(18,·)$, $\chi_{47}(19,·)$, $\chi_{47}(20,·)$, $\chi_{47}(21,·)$, $\chi_{47}(22,·)$, $\chi_{47}(23,·)$, $\chi_{47}(24,·)$, $\chi_{47}(25,·)$, $\chi_{47}(26,·)$, $\chi_{47}(27,·)$, $\chi_{47}(28,·)$, $\chi_{47}(29,·)$, $\chi_{47}(30,·)$, $\chi_{47}(31,·)$, $\chi_{47}(32,·)$, $\chi_{47}(33,·)$, $\chi_{47}(34,·)$, $\chi_{47}(35,·)$, $\chi_{47}(36,·)$, $\chi_{47}(37,·)$, $\chi_{47}(38,·)$, $\chi_{47}(39,·)$, $\chi_{47}(40,·)$, $\chi_{47}(41,·)$, $\chi_{47}(42,·)$, $\chi_{47}(43,·)$, $\chi_{47}(44,·)$, $\chi_{47}(45,·)$, $\chi_{47}(46,·)$$\rbrace$
This is a CM field.

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

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

Class group and class number

$C_{695}$, which has order $695$ (assuming GRH)

Unit group

Rank:		$22$
Torsion generator:		\( a \) (order $94$)
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:		\( 286117566502019100 \) (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

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

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

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
47	Data not computed