Global number field 32.32.62912853223226562597999842165869507304166444172308381696.1: \(\Q(\zeta_{192})^+\)

• Label: 32.32.6291285322...1696.1
• Degree: $32$
• Signature: $[32, 0]$
• Discriminant: $2^{160}\cdot 3^{16}$
• Root discriminant: $55.43$
• Ramified primes: $2, 3$
• Class number: $1$ (GRH)
• Class group: Trivial (GRH)
• Galois group: $C_2\times C_{16}$ (as 32T32)

Normalized defining polynomial

\( x^{32} - 32 x^{30} + 464 x^{28} - 4032 x^{26} + 23400 x^{24} - 95680 x^{22} + 283360 x^{20} - 615296 x^{18} + 980628 x^{16} - 1136960 x^{14} + 940576 x^{12} - 537472 x^{10} + 201552 x^{8} - 45696 x^{6} + 5440 x^{4} - 256 x^{2} + 1 \)

Invariants

Degree:		$32$
Signature:		$[32, 0]$
Discriminant:		\(62912853223226562597999842165869507304166444172308381696=2^{160}\cdot 3^{16}\)
Root discriminant:		$55.43$
Ramified primes:		$2, 3$
This field is Galois and abelian over $\Q$.
Conductor:		\(192=2^{6}\cdot 3\)
Dirichlet character group:		$\lbrace$$\chi_{192}(1,·)$, $\chi_{192}(131,·)$, $\chi_{192}(133,·)$, $\chi_{192}(11,·)$, $\chi_{192}(13,·)$, $\chi_{192}(143,·)$, $\chi_{192}(145,·)$, $\chi_{192}(23,·)$, $\chi_{192}(25,·)$, $\chi_{192}(155,·)$, $\chi_{192}(157,·)$, $\chi_{192}(35,·)$, $\chi_{192}(37,·)$, $\chi_{192}(167,·)$, $\chi_{192}(169,·)$, $\chi_{192}(47,·)$, $\chi_{192}(49,·)$, $\chi_{192}(179,·)$, $\chi_{192}(181,·)$, $\chi_{192}(59,·)$, $\chi_{192}(61,·)$, $\chi_{192}(191,·)$, $\chi_{192}(71,·)$, $\chi_{192}(73,·)$, $\chi_{192}(83,·)$, $\chi_{192}(85,·)$, $\chi_{192}(95,·)$, $\chi_{192}(97,·)$, $\chi_{192}(107,·)$, $\chi_{192}(109,·)$, $\chi_{192}(119,·)$, $\chi_{192}(121,·)$$\rbrace$
This is not a CM field.

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

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

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:		\( 969667117813122800 \) (assuming GRH)

Galois group

$C_2\times C_{16}$ (as 32T32):

An abelian group of order 32

The 32 conjugacy class representatives for $C_2\times C_{16}$

Character table for $C_2\times C_{16}$ is not computed

Intermediate fields

\(\Q(\sqrt{6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\zeta_{16})^+\), 4.4.18432.1, \(\Q(\zeta_{48})^+\), \(\Q(\zeta_{32})^+\), 8.8.173946175488.1, \(\Q(\zeta_{96})^+\), 16.16.3965881151245791007623610368.1, \(\Q(\zeta_{64})^+\)

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	R	$16^{2}$	${\href{/LocalNumberField/7.8.0.1}{8} }^{4}$	$16^{2}$	$16^{2}$	${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$	$16^{2}$	${\href{/LocalNumberField/23.8.0.1}{8} }^{4}$	$16^{2}$	${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$	$16^{2}$	${\href{/LocalNumberField/41.8.0.1}{8} }^{4}$	$16^{2}$	${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$	$16^{2}$	$16^{2}$

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
2	Data not computed
3	Data not computed