Global number field 32.0.151142765320815856472639544599622796222554813389533609984.2

• Label: 32.0.15114276532...9984.2
• Degree: $32$
• Signature: $[0, 16]$
• Discriminant: $2^{64}\cdot 17^{30}$
• Root discriminant: $56.96$
• Ramified primes: $2, 17$
• Class number: $3104$ (GRH)
• Class group: $[2, 2, 776]$ (GRH)
• Galois group: $C_2\times C_{16}$ (as 32T32)

Normalized defining polynomial

\( x^{32} + 51 x^{28} + 1003 x^{24} + 9554 x^{20} + 45407 x^{16} + 99994 x^{12} + 83810 x^{8} + 17340 x^{4} + 289 \)

Invariants

Degree:		$32$
Signature:		$[0, 16]$
Discriminant:		\(151142765320815856472639544599622796222554813389533609984=2^{64}\cdot 17^{30}\)
Root discriminant:		$56.96$
Ramified primes:		$2, 17$
This field is Galois and abelian over $\Q$.
Conductor:		\(136=2^{3}\cdot 17\)
Dirichlet character group:		$\lbrace$$\chi_{136}(1,·)$, $\chi_{136}(3,·)$, $\chi_{136}(5,·)$, $\chi_{136}(135,·)$, $\chi_{136}(9,·)$, $\chi_{136}(11,·)$, $\chi_{136}(15,·)$, $\chi_{136}(131,·)$, $\chi_{136}(25,·)$, $\chi_{136}(27,·)$, $\chi_{136}(29,·)$, $\chi_{136}(133,·)$, $\chi_{136}(33,·)$, $\chi_{136}(37,·)$, $\chi_{136}(45,·)$, $\chi_{136}(47,·)$, $\chi_{136}(49,·)$, $\chi_{136}(55,·)$, $\chi_{136}(61,·)$, $\chi_{136}(75,·)$, $\chi_{136}(81,·)$, $\chi_{136}(87,·)$, $\chi_{136}(89,·)$, $\chi_{136}(91,·)$, $\chi_{136}(99,·)$, $\chi_{136}(103,·)$, $\chi_{136}(107,·)$, $\chi_{136}(109,·)$, $\chi_{136}(111,·)$, $\chi_{136}(121,·)$, $\chi_{136}(125,·)$, $\chi_{136}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{17} a^{16}$, $\frac{1}{17} a^{17}$, $\frac{1}{17} a^{18}$, $\frac{1}{17} a^{19}$, $\frac{1}{17} a^{20}$, $\frac{1}{17} a^{21}$, $\frac{1}{17} a^{22}$, $\frac{1}{17} a^{23}$, $\frac{1}{17} a^{24}$, $\frac{1}{17} a^{25}$, $\frac{1}{17} a^{26}$, $\frac{1}{17} a^{27}$, $\frac{1}{2638828420553} a^{28} + \frac{22203801691}{2638828420553} a^{24} + \frac{39662487090}{2638828420553} a^{20} - \frac{3473372063}{2638828420553} a^{16} - \frac{74276387048}{155225201209} a^{12} - \frac{36448458563}{155225201209} a^{8} + \frac{12215456199}{155225201209} a^{4} - \frac{1308144924}{155225201209}$, $\frac{1}{2638828420553} a^{29} + \frac{22203801691}{2638828420553} a^{25} + \frac{39662487090}{2638828420553} a^{21} - \frac{3473372063}{2638828420553} a^{17} - \frac{74276387048}{155225201209} a^{13} - \frac{36448458563}{155225201209} a^{9} + \frac{12215456199}{155225201209} a^{5} - \frac{1308144924}{155225201209} a$, $\frac{1}{44860083149401} a^{30} - \frac{44348364962}{2638828420553} a^{26} + \frac{75380240986}{2638828420553} a^{22} - \frac{27596998570}{2638828420553} a^{18} - \frac{850402393093}{2638828420553} a^{14} + \frac{25248655592}{155225201209} a^{10} + \frac{718556247}{155225201209} a^{6} - \frac{54862314834}{155225201209} a^{2}$, $\frac{1}{44860083149401} a^{31} - \frac{44348364962}{2638828420553} a^{27} + \frac{75380240986}{2638828420553} a^{23} - \frac{27596998570}{2638828420553} a^{19} - \frac{850402393093}{2638828420553} a^{15} + \frac{25248655592}{155225201209} a^{11} + \frac{718556247}{155225201209} a^{7} - \frac{54862314834}{155225201209} a^{3}$

Class group and class number

$C_{2}\times C_{2}\times C_{776}$, which has order $3104$ (assuming GRH)

Unit group

Rank:		$15$
Torsion generator:		\( -\frac{15659481977}{44860083149401} a^{30} - \frac{47011167184}{2638828420553} a^{26} - \frac{925587714696}{2638828420553} a^{22} - \frac{8833942592480}{2638828420553} a^{18} - \frac{42149161816084}{2638828420553} a^{14} - \frac{5511608578720}{155225201209} a^{10} - \frac{4767220925456}{155225201209} a^{6} - \frac{1234478842176}{155225201209} a^{2} \) (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:		\( 598124168304.8442 \) (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, 16.0.48023489818559305679372288.1, 16.16.48023489818559305679372288.1

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	$16^{2}$	$16^{2}$	$16^{2}$	$16^{2}$	${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$	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
17	Data not computed