Number field 32.0.3138550867693340381917894711603833208051177722232017256448.1

• Label: 32.0.313...448.1
• Degree: $32$
• Signature: $[0, 16]$
• Discriminant: $3.139\times 10^{57}$
• Root discriminant: \(62.63\)
• Ramified prime: $2$
• Class number: $21121$ (GRH)
• Class group: [21121] (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:		$[0, 16]$
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$$\chi_{128}(1,·)$, $\chi_{128}(3,·)$, $\chi_{128}(9,·)$, $\chi_{128}(11,·)$, $\chi_{128}(17,·)$, $\chi_{128}(19,·)$, $\chi_{128}(25,·)$, $\chi_{128}(27,·)$, $\chi_{128}(33,·)$, $\chi_{128}(35,·)$, $\chi_{128}(41,·)$, $\chi_{128}(43,·)$, $\chi_{128}(49,·)$, $\chi_{128}(51,·)$, $\chi_{128}(57,·)$, $\chi_{128}(59,·)$, $\chi_{128}(65,·)$, $\chi_{128}(67,·)$, $\chi_{128}(73,·)$, $\chi_{128}(75,·)$, $\chi_{128}(81,·)$, $\chi_{128}(83,·)$, $\chi_{128}(89,·)$, $\chi_{128}(91,·)$, $\chi_{128}(97,·)$, $\chi_{128}(99,·)$, $\chi_{128}(105,·)$, $\chi_{128}(107,·)$, $\chi_{128}(113,·)$, $\chi_{128}(115,·)$, $\chi_{128}(121,·)$, $\chi_{128}(123,·)$$\rbrace$
This is a CM field.
Reflex fields:		unavailable$^{32768}$

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

$C_{21121}$, which has order $21121$ (assuming GRH)

Unit group

Rank:		$15$
Torsion generator:		\( -1 \)  (order $2$)
Fundamental units:		$a^{20}+20a^{18}+170a^{16}+800a^{14}+2275a^{12}+4004a^{10}+4290a^{8}+2640a^{6}+825a^{4}+100a^{2}+3$, $a^{10}+10a^{8}+35a^{6}+50a^{4}+25a^{2}+3$, $a^{4}+4a^{2}+1$, $a^{30}+30a^{28}+405a^{26}+3250a^{24}+17250a^{22}+63756a^{20}+168245a^{18}+319770a^{16}+436050a^{14}+419900a^{12}+277134a^{10}+119340a^{8}+30940a^{6}+4200a^{4}+225a^{2}+1$, $a^{18}+18a^{16}+135a^{14}+546a^{12}+1287a^{10}+1782a^{8}+1386a^{6}+540a^{4}+81a^{2}+1$, $a^{30}+31a^{28}+432a^{26}+3573a^{24}+19503a^{22}+73900a^{20}+199065a^{18}+383911a^{16}+526932a^{14}+505257a^{12}+327471a^{10}+136135a^{8}+33397a^{6}+4179a^{4}+188a^{2}-1$, $a^{26}+26a^{24}+299a^{22}+2002a^{20}+8645a^{18}+25194a^{16}+50388a^{14}+68952a^{12}+63206a^{10}+37180a^{8}+13013a^{6}+2366a^{4}+169a^{2}+3$, $a^{26}+26a^{24}+299a^{22}+2002a^{20}+8645a^{18}+25194a^{16}+50388a^{14}+68952a^{12}+63206a^{10}+37180a^{8}+13013a^{6}+2366a^{4}+169a^{2}+1$, $a^{28}+28a^{26}+350a^{24}+2576a^{22}+12397a^{20}+40964a^{18}+94962a^{16}+155040a^{14}+176358a^{12}+136136a^{10}+68068a^{8}+20384a^{6}+3185a^{4}+196a^{2}+1$, $a^{24}+24a^{22}+252a^{20}+1520a^{18}+5814a^{16}+14688a^{14}+24752a^{12}+27456a^{10}+19305a^{8}+8008a^{6}+1716a^{4}+144a^{2}+3$, $a^{24}+24a^{22}+252a^{20}+1520a^{18}+5814a^{16}+14688a^{14}+24752a^{12}+27456a^{10}+19305a^{8}+8008a^{6}+1716a^{4}+144a^{2}+1$, $a^{8}+8a^{6}+20a^{4}+16a^{2}+1$, $a^{30}+30a^{28}+405a^{26}+3250a^{24}+17250a^{22}+63757a^{20}+168265a^{18}+319940a^{16}+436850a^{14}+422175a^{12}+281139a^{10}+123640a^{8}+33615a^{6}+5075a^{4}+350a^{2}+7$, $a^{30}+30a^{28}+405a^{26}+3250a^{24}+17250a^{22}+63756a^{20}+168245a^{18}+319770a^{16}+436050a^{14}+419900a^{12}+277134a^{10}+119340a^{8}+30940a^{6}+4199a^{4}+221a^{2}+1$, $a^{26}+26a^{24}+299a^{22}+2002a^{20}+8645a^{18}+25194a^{16}+50388a^{14}+68953a^{12}+63218a^{10}+37234a^{8}+13125a^{6}+2471a^{4}+205a^{2}+3$ (assuming GRH)
Regulator:		\( 211230625393.46567 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 211230625393.46567 \cdot 21121}{2\cdot\sqrt{3138550867693340381917894711603833208051177722232017256448}}\cr\approx \mathstrut & 0.234938760729838 \end{aligned}\] (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 $32$	$32$	$1$	$191$