Global number field 46.46.123533119255810717326269698346313186386460569318540167792790808431593601417039621597954048.1: \(\Q(\zeta_{188})^+\)

• Label: 46.46.1235331192...4048.1
• Degree: $46$
• Signature: $[46, 0]$
• Discriminant: $2^{46}\cdot 47^{45}$
• Root discriminant: $86.45$
• Ramified primes: $2, 47$
• Class number: Not computed
• Class group: Not computed
• Galois group: $C_{46}$ (as 46T1)

Normalized defining polynomial

\( x^{46} - 47 x^{44} + 1034 x^{42} - 14147 x^{40} + 134890 x^{38} - 951938 x^{36} + 5154396 x^{34} - 21906183 x^{32} + 74144004 x^{30} - 201619660 x^{28} + 442473416 x^{26} - 784384692 x^{24} + 1120549560 x^{22} - 1282801080 x^{20} + 1166182800 x^{18} - 830905245 x^{16} + 455657715 x^{14} - 187623765 x^{12} + 56071470 x^{10} - 11593725 x^{8} + 1545830 x^{6} - 118910 x^{4} + 4324 x^{2} - 47 \)

Invariants

Degree:		$46$
Signature:		$[46, 0]$
Discriminant:		\(123533119255810717326269698346313186386460569318540167792790808431593601417039621597954048=2^{46}\cdot 47^{45}\)
Root discriminant:		$86.45$
Ramified primes:		$2, 47$
This field is Galois and abelian over $\Q$.
Conductor:		\(188=2^{2}\cdot 47\)
Dirichlet character group:		$\lbrace$$\chi_{188}(1,·)$, $\chi_{188}(43,·)$, $\chi_{188}(135,·)$, $\chi_{188}(9,·)$, $\chi_{188}(11,·)$, $\chi_{188}(15,·)$, $\chi_{188}(17,·)$, $\chi_{188}(19,·)$, $\chi_{188}(149,·)$, $\chi_{188}(23,·)$, $\chi_{188}(127,·)$, $\chi_{188}(25,·)$, $\chi_{188}(157,·)$, $\chi_{188}(31,·)$, $\chi_{188}(35,·)$, $\chi_{188}(37,·)$, $\chi_{188}(167,·)$, $\chi_{188}(169,·)$, $\chi_{188}(171,·)$, $\chi_{188}(173,·)$, $\chi_{188}(49,·)$, $\chi_{188}(179,·)$, $\chi_{188}(53,·)$, $\chi_{188}(187,·)$, $\chi_{188}(151,·)$, $\chi_{188}(61,·)$, $\chi_{188}(65,·)$, $\chi_{188}(67,·)$, $\chi_{188}(177,·)$, $\chi_{188}(81,·)$, $\chi_{188}(163,·)$, $\chi_{188}(139,·)$, $\chi_{188}(87,·)$, $\chi_{188}(89,·)$, $\chi_{188}(91,·)$, $\chi_{188}(165,·)$, $\chi_{188}(97,·)$, $\chi_{188}(99,·)$, $\chi_{188}(101,·)$, $\chi_{188}(145,·)$, $\chi_{188}(107,·)$, $\chi_{188}(153,·)$, $\chi_{188}(121,·)$, $\chi_{188}(123,·)$, $\chi_{188}(39,·)$, $\chi_{188}(21,·)$$\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

Not computed

Unit group

Rank:		$45$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		Not computed
Regulator:		Not computed

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

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
2	Data not computed
47	Data not computed