Global number field 32.0.1461501637330902918203684832716283019655932542976.1: \(\Q(\zeta_{64})\)

• Label: 32.0.14615016373...2976.1
• Degree: $32$
• Signature: $[0, 16]$
• Discriminant: $2^{160}$
• Root discriminant: $32.00$
• Ramified prime: $2$
• Class number: $17$ (GRH)
• Class group: $[17]$ (GRH)
• Galois group: $C_2\times C_{16}$ (as 32T32)

Normalized defining polynomial

\( x^{32} + 1 \)

Invariants

Degree:		$32$
Signature:		$[0, 16]$
Discriminant:		\(1461501637330902918203684832716283019655932542976=2^{160}\)
Root discriminant:		$32.00$
Ramified primes:		$2$
This field is Galois and abelian over $\Q$.
Conductor:		\(64=2^{6}\)
Dirichlet character group:		$\lbrace$$\chi_{64}(1,·)$, $\chi_{64}(3,·)$, $\chi_{64}(5,·)$, $\chi_{64}(7,·)$, $\chi_{64}(9,·)$, $\chi_{64}(11,·)$, $\chi_{64}(13,·)$, $\chi_{64}(15,·)$, $\chi_{64}(17,·)$, $\chi_{64}(19,·)$, $\chi_{64}(21,·)$, $\chi_{64}(23,·)$, $\chi_{64}(25,·)$, $\chi_{64}(27,·)$, $\chi_{64}(29,·)$, $\chi_{64}(31,·)$, $\chi_{64}(33,·)$, $\chi_{64}(35,·)$, $\chi_{64}(37,·)$, $\chi_{64}(39,·)$, $\chi_{64}(41,·)$, $\chi_{64}(43,·)$, $\chi_{64}(45,·)$, $\chi_{64}(47,·)$, $\chi_{64}(49,·)$, $\chi_{64}(51,·)$, $\chi_{64}(53,·)$, $\chi_{64}(55,·)$, $\chi_{64}(57,·)$, $\chi_{64}(59,·)$, $\chi_{64}(61,·)$, $\chi_{64}(63,·)$$\rbrace$
This is a CM field.

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

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

Class group and class number

$C_{17}$, which has order $17$ (assuming GRH)

Unit group

Rank:		$15$
Torsion generator:		\( a \) (order $64$)
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:		\( 211230625393.46567 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\zeta_{8})\), \(\Q(\zeta_{16})^+\), 4.0.2048.2, \(\Q(\zeta_{16})\), \(\Q(\zeta_{32})^+\), 8.0.2147483648.1, \(\Q(\zeta_{32})\), \(\Q(\zeta_{64})^+\), 16.0.604462909807314587353088.1

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

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