Global number field 32.32.3138550867693340381917894711603833208051177722232017256448.1: \(\Q(\zeta_{128})^+\)

• Label: 32.32.3138550867...6448.1
• Degree: $32$
• Signature: $[32, 0]$
• Discriminant: $2^{191}$
• Root discriminant: $62.63$
• Ramified prime: $2$
• Class number: $1$ (GRH)
• Class group: Trivial (GRH)
• Galois group: $C_{32}$ (as 32T33)

Normalized defining polynomial

\( x^{32} - 32 x^{30} + 464 x^{28} - 4032 x^{26} + 23400 x^{24} - 95680 x^{22} + 283360 x^{20} - 615296 x^{18} + 980628 x^{16} - 1136960 x^{14} + 940576 x^{12} - 537472 x^{10} + 201552 x^{8} - 45696 x^{6} + 5440 x^{4} - 256 x^{2} + 2 \)

Invariants

Degree:		$32$
Signature:		$[32, 0]$
Discriminant:		\(3138550867693340381917894711603833208051177722232017256448=2^{191}\)
Root discriminant:		$62.63$
Ramified primes:		$2$
This field is Galois and abelian over $\Q$.
Conductor:		\(128=2^{7}\)
Dirichlet character group:		$\lbrace$$\chi_{128}(1,·)$, $\chi_{128}(5,·)$, $\chi_{128}(9,·)$, $\chi_{128}(13,·)$, $\chi_{128}(17,·)$, $\chi_{128}(21,·)$, $\chi_{128}(25,·)$, $\chi_{128}(29,·)$, $\chi_{128}(33,·)$, $\chi_{128}(37,·)$, $\chi_{128}(41,·)$, $\chi_{128}(45,·)$, $\chi_{128}(49,·)$, $\chi_{128}(53,·)$, $\chi_{128}(57,·)$, $\chi_{128}(61,·)$, $\chi_{128}(65,·)$, $\chi_{128}(69,·)$, $\chi_{128}(73,·)$, $\chi_{128}(77,·)$, $\chi_{128}(81,·)$, $\chi_{128}(85,·)$, $\chi_{128}(89,·)$, $\chi_{128}(93,·)$, $\chi_{128}(97,·)$, $\chi_{128}(101,·)$, $\chi_{128}(105,·)$, $\chi_{128}(109,·)$, $\chi_{128}(113,·)$, $\chi_{128}(117,·)$, $\chi_{128}(121,·)$, $\chi_{128}(125,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$

Class group and class number

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

Unit group

Rank:		$31$
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:		\( 7254970870890416000 \) (assuming GRH)

Galois group

$C_{32}$ (as 32T33):

A cyclic group of order 32

The 32 conjugacy class representatives for $C_{32}$

Character table for $C_{32}$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), \(\Q(\zeta_{32})^+\), \(\Q(\zeta_{64})^+\)

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	$32$	$32$	$16^{2}$	$32$	$32$	${\href{/LocalNumberField/17.8.0.1}{8} }^{4}$	$32$	$16^{2}$	$32$	${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$	$32$	$16^{2}$	$32$	${\href{/LocalNumberField/47.8.0.1}{8} }^{4}$	$32$	$32$

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