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

• Label: 32.32.313...448.1
• Degree: $32$
• Signature: $[32, 0]$
• Discriminant: $3.139\times 10^{57}$
• 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:		\(3138550867693340381917894711603833208051177722232017256448\) \(\medspace = 2^{191}\)
Root discriminant:		\(62.63\)
Galois root discriminant:		$2^{191/32}\approx 62.628611973612806$
Ramified primes:		\(2\)
Discriminant root field:		\(\Q(\sqrt{2}) \)
$\card{ \Gal(K/\Q) }$:		$32$
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 \(a\))

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

Monogenic:		Yes
Index:		$1$
Inessential primes:		None

Class group and class number

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

Unit group

Rank:		$31$
Torsion generator:		\( -1 \)  (order $2$)
Fundamental units:		$a^{16}-16a^{14}+104a^{12}-352a^{10}+660a^{8}-672a^{6}+336a^{4}-64a^{2}+1$, $a^{8}-8a^{6}+20a^{4}-16a^{2}+3$, $a^{24}-24a^{22}+252a^{20}-1520a^{18}+5814a^{16}-14688a^{14}+24752a^{12}-27456a^{10}+19305a^{8}-8008a^{6}+1716a^{4}-144a^{2}+1$, $a^{20}-20a^{18}+170a^{16}-800a^{14}+2275a^{12}-4004a^{10}+4290a^{8}-2640a^{6}+825a^{4}-100a^{2}+3$, $a^{4}-4a^{2}+3$, $a^{28}-28a^{26}+350a^{24}-2576a^{22}+12396a^{20}-40944a^{18}+94792a^{16}-154240a^{14}+174083a^{12}-132132a^{10}+63779a^{8}-17752a^{6}+2380a^{4}-112a^{2}+1$, $a^{28}-28a^{26}+350a^{24}-2576a^{22}+12397a^{20}-40964a^{18}+94962a^{16}-155040a^{14}+176358a^{12}-136136a^{10}+68068a^{8}-20384a^{6}+3185a^{4}-196a^{2}+3$, $a^{22}-22a^{20}+209a^{18}-1122a^{16}+3740a^{14}-8008a^{12}+11011a^{10}-9438a^{8}+4719a^{6}-1210a^{4}+121a^{2}-3$, $a^{14}-14a^{12}+77a^{10}-210a^{8}+294a^{6}-196a^{4}+49a^{2}-1$, $a^{2}-3$, $a^{28}-28a^{26}+350a^{24}-2576a^{22}+12397a^{20}-40963a^{18}+94944a^{16}-154905a^{14}+175812a^{12}-134850a^{10}+66296a^{8}-19033a^{6}+2695a^{4}-140a^{2}+1$, $a^{26}-26a^{24}+299a^{22}-2002a^{20}+8645a^{18}-25194a^{16}+50388a^{14}-68952a^{12}+63206a^{10}-37180a^{8}+13013a^{6}-2366a^{4}+169a^{2}-1$, $a^{28}-28a^{26}+350a^{24}-2576a^{22}+12397a^{20}-40964a^{18}+94962a^{16}-155039a^{14}+176344a^{12}-136059a^{10}+67858a^{8}-20090a^{6}+2989a^{4}-147a^{2}+1$, $a^{20}-20a^{18}+170a^{16}-800a^{14}+2275a^{12}-4005a^{10}+4300a^{8}-2675a^{6}+875a^{4}-125a^{2}+5$, $a^{14}-13a^{12}+66a^{10}-165a^{8}+210a^{6}-126a^{4}+28a^{2}-1$, $a^{25}-25a^{23}+275a^{21}-1750a^{19}+7125a^{17}-19380a^{15}+35700a^{13}-44200a^{11}+35750a^{9}-17875a^{7}+5005a^{5}-650a^{3}+25a+1$, $a^{11}-11a^{9}+44a^{7}-77a^{5}+55a^{3}-11a+1$, $a^{31}-31a^{29}+434a^{27}-3627a^{25}+20150a^{23}-78430a^{21}+219604a^{19}-447051a^{17}+660858a^{15}-700910a^{13}+520676a^{11}-260338a^{9}+82212a^{7}-14756a^{5}+1240a^{3}-31a-1$, $a^{25}-25a^{23}+275a^{21}-1750a^{19}+7125a^{17}-19380a^{15}-a^{14}+35700a^{13}+14a^{12}-44201a^{11}-77a^{10}+35761a^{9}+210a^{8}-17919a^{7}-294a^{6}+5082a^{5}+196a^{4}-705a^{3}-49a^{2}+36a+3$, $a^{31}-31a^{29}+434a^{27}-3627a^{25}+20150a^{23}+a^{22}-78430a^{21}-23a^{20}+219604a^{19}+229a^{18}-447051a^{17}-1292a^{16}+660858a^{15}+4540a^{14}-700909a^{13}-10283a^{12}+520662a^{11}+15015a^{10}-260261a^{9}-13728a^{8}+82003a^{7}+7359a^{6}-14470a^{5}-2035a^{4}+1064a^{3}+220a^{2}+2a-1$, $a^{29}-29a^{27}+377a^{25}-2901a^{23}+14697a^{21}-51589a^{19}+128592a^{17}-229500a^{15}+291958a^{13}-260286a^{11}+157443a^{9}-61139a^{7}+a^{6}+13845a^{5}-6a^{4}-1521a^{3}+9a^{2}+52a-3$, $a^{13}-13a^{11}+65a^{9}-156a^{7}+182a^{5}-91a^{3}+13a-1$, $a^{29}-29a^{27}+377a^{25}-2900a^{23}+14674a^{21}-51359a^{19}+127281a^{17}-224808a^{15}+281010a^{13}-243542a^{11}+140998a^{9}-51272a^{7}+10556a^{5}-1015a^{3}+29a-1$, $a^{10}-10a^{8}+35a^{6}+a^{5}-50a^{4}-5a^{3}+25a^{2}+5a-1$, $a^{30}-30a^{28}+405a^{26}-3250a^{24}+17250a^{22}-63756a^{20}+168245a^{18}-319770a^{16}-a^{15}+436050a^{14}+15a^{13}-419900a^{12}-90a^{11}+277134a^{10}+275a^{9}-119340a^{8}-450a^{7}+30940a^{6}+378a^{5}-4200a^{4}-140a^{3}+225a^{2}+15a-1$, $a^{31}-32a^{29}+463a^{27}-4004a^{25}-a^{24}+23050a^{23}+25a^{22}-93104a^{21}-275a^{20}+270963a^{19}+1750a^{18}-574332a^{17}-7124a^{16}+885666a^{15}+19363a^{14}-981919a^{13}-35581a^{12}+764204a^{11}+43758a^{10}-401259a^{9}-34815a^{8}+133274a^{7}+16753a^{6}-25019a^{5}-4290a^{4}+2065a^{3}+441a^{2}-20a-3$, $a^{31}-32a^{29}+464a^{27}-4031a^{25}+23374a^{23}-a^{22}-95381a^{21}+23a^{20}+281358a^{19}-229a^{18}-606651a^{17}+1292a^{16}+955433a^{15}-4540a^{14}-1086556a^{13}+10283a^{12}+871520a^{11}-15015a^{10}-473914a^{9}+13728a^{8}+163712a^{7}-7359a^{6}-32012a^{5}+2035a^{4}+2744a^{3}-221a^{2}-32a+3$, $a^{23}+a^{22}-22a^{21}-21a^{20}+209a^{19}+189a^{18}-1122a^{17}-952a^{16}+3740a^{15}+2940a^{14}-8008a^{13}-5733a^{12}+11011a^{11}+7007a^{10}-9438a^{9}-5148a^{8}+4719a^{7}+2079a^{6}-1210a^{5}-385a^{4}+121a^{3}+21a^{2}-3a-1$, $a^{22}-22a^{20}+209a^{18}-1122a^{16}+3740a^{14}-8008a^{12}+a^{11}+11011a^{10}-11a^{9}-9438a^{8}+44a^{7}+4719a^{6}-77a^{5}-1210a^{4}+55a^{3}+121a^{2}-11a-1$, $a^{12}-a^{11}-11a^{10}+10a^{9}+45a^{8}-36a^{7}-84a^{6}+56a^{5}+70a^{4}-35a^{3}-21a^{2}+6a+1$, $a^{25}-25a^{23}+275a^{21}-1750a^{19}+7125a^{17}-19380a^{15}+a^{14}+35700a^{13}-14a^{12}-44200a^{11}+77a^{10}+35750a^{9}-210a^{8}-17875a^{7}+294a^{6}+5005a^{5}-196a^{4}-650a^{3}+49a^{2}+25a-3$ (assuming GRH)
Regulator:		\( 7254970870890416000 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{32}\cdot(2\pi)^{0}\cdot 7254970870890416000 \cdot 1}{2\cdot\sqrt{3138550867693340381917894711603833208051177722232017256448}}\cr\approx \mathstrut & 0.278099976864148 \end{aligned}\] (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	$32$	$32$	$16^{2}$	$32$	$32$	${\href{/padicField/17.8.0.1}{8} }^{4}$	$32$	$16^{2}$	$32$	${\href{/padicField/31.4.0.1}{4} }^{8}$	$32$	$16^{2}$	$32$	${\href{/padicField/47.8.0.1}{8} }^{4}$	$32$	$32$

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
\(2\)		Deg $32$	$32$	$1$	$191$