Global number field 32.0.1514842838499144573219529960757824409341479409296605184.1

• Label: 32.0.15148428384...5184.1
• Degree: $32$
• Signature: $[0, 16]$
• Discriminant: $2^{32}\cdot 3^{16}\cdot 17^{30}$
• Root discriminant: $49.33$
• Ramified primes: $2, 3, 17$
• Class number: $6176$ (GRH)
• Class group: $[2, 3088]$ (GRH)
• Galois group: $C_2\times C_{16}$ (as 32T32)

Normalized defining polynomial

\( x^{32} + 33 x^{30} + 494 x^{28} + 4436 x^{26} + 26623 x^{24} + 112607 x^{22} + 344863 x^{20} + 773397 x^{18} + 1269578 x^{16} + 1508868 x^{14} + 1269578 x^{12} + 729146 x^{10} + 270216 x^{8} + 59244 x^{6} + 6648 x^{4} + 288 x^{2} + 1 \)

Invariants

Degree:		$32$
Signature:		$[0, 16]$
Discriminant:		\(1514842838499144573219529960757824409341479409296605184=2^{32}\cdot 3^{16}\cdot 17^{30}\)
Root discriminant:		$49.33$
Ramified primes:		$2, 3, 17$
This field is Galois and abelian over $\Q$.
Conductor:		\(204=2^{2}\cdot 3\cdot 17\)
Dirichlet character group:		$\lbrace$$\chi_{204}(1,·)$, $\chi_{204}(131,·)$, $\chi_{204}(5,·)$, $\chi_{204}(11,·)$, $\chi_{204}(13,·)$, $\chi_{204}(143,·)$, $\chi_{204}(145,·)$, $\chi_{204}(19,·)$, $\chi_{204}(23,·)$, $\chi_{204}(25,·)$, $\chi_{204}(157,·)$, $\chi_{204}(167,·)$, $\chi_{204}(41,·)$, $\chi_{204}(43,·)$, $\chi_{204}(173,·)$, $\chi_{204}(29,·)$, $\chi_{204}(49,·)$, $\chi_{204}(55,·)$, $\chi_{204}(151,·)$, $\chi_{204}(65,·)$, $\chi_{204}(67,·)$, $\chi_{204}(197,·)$, $\chi_{204}(71,·)$, $\chi_{204}(95,·)$, $\chi_{204}(103,·)$, $\chi_{204}(107,·)$, $\chi_{204}(113,·)$, $\chi_{204}(115,·)$, $\chi_{204}(169,·)$, $\chi_{204}(121,·)$, $\chi_{204}(125,·)$, $\chi_{204}(127,·)$$\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

$C_{2}\times C_{3088}$, which has order $6176$ (assuming GRH)

Unit group

Rank:		$15$
Torsion generator:		\( a^{17} + 17 a^{15} + 119 a^{13} + 442 a^{11} + 935 a^{9} + 1122 a^{7} + 714 a^{5} + 204 a^{3} + 17 a \) (order $4$)
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:		\( 19955290291.92932 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-17}) \), \(\Q(i, \sqrt{17})\), 4.4.4913.1, 4.0.78608.1, 8.0.6179217664.1, \(\Q(\zeta_{17})^+\), 8.0.105046700288.1, 16.0.11034809241396899282944.1, \(\Q(\zeta_{51})^+\), 16.0.1230789518357685955321724928.1

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	R	$16^{2}$	$16^{2}$	$16^{2}$	${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$	R	${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$	$16^{2}$	$16^{2}$	$16^{2}$	$16^{2}$	$16^{2}$	${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$	${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$	${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$	${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

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
17	Data not computed