Number field 23.23.6111571184724799803076702357055363809.1: \(\Q(\zeta_{47})^+\)

• Label: 23.23.611...809.1
• Degree: $23$
• Signature: $[23, 0]$
• Discriminant: $6.112\times 10^{36}$
• Root discriminant: \(39.76\)
• Ramified prime: $47$
• Class number: $1$ (GRH)
• Class group: trivial (GRH)
• Galois group: $C_{23}$ (as 23T1)

Normalized defining polynomial

x^23 - x^22 - 22*x^21 + 21*x^20 + 210*x^19 - 190*x^18 - 1140*x^17 + 969*x^16 + 3876*x^15 - 3060*x^14 - 8568*x^13 + 6188*x^12 + 12376*x^11 - 8008*x^10 - 11440*x^9 + 6435*x^8 + 6435*x^7 - 3003*x^6 - 2002*x^5 + 715*x^4 + 286*x^3 - 66*x^2 - 12*x + 1

Invariants

Degree:		$23$
Signature:		$[23, 0]$
Discriminant:		\(6111571184724799803076702357055363809\) \(\medspace = 47^{22}\)
Root discriminant:		\(39.76\)
Galois root discriminant:		$47^{22/23}\approx 39.75556693193454$
Ramified primes:		\(47\)
Discriminant root field:		\(\Q\)
$\card{ \Gal(K/\Q) }$:		$23$
This field is Galois and abelian over $\Q$.
Conductor:		\(47\)
Dirichlet character group:		$\lbrace$$\chi_{47}(1,·)$, $\chi_{47}(2,·)$, $\chi_{47}(3,·)$, $\chi_{47}(4,·)$, $\chi_{47}(6,·)$, $\chi_{47}(7,·)$, $\chi_{47}(8,·)$, $\chi_{47}(9,·)$, $\chi_{47}(12,·)$, $\chi_{47}(14,·)$, $\chi_{47}(16,·)$, $\chi_{47}(17,·)$, $\chi_{47}(18,·)$, $\chi_{47}(21,·)$, $\chi_{47}(24,·)$, $\chi_{47}(25,·)$, $\chi_{47}(27,·)$, $\chi_{47}(28,·)$, $\chi_{47}(32,·)$, $\chi_{47}(34,·)$, $\chi_{47}(36,·)$, $\chi_{47}(37,·)$, $\chi_{47}(42,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$

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

Class group and class number

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

Unit group

Rank:		$22$
Torsion generator:		\( -1 \)  (order $2$)
Fundamental units:		$a^{22}-21a^{20}+190a^{18}-969a^{16}+3060a^{14}-6188a^{12}+8008a^{10}-6435a^{8}+3003a^{6}-715a^{4}+66a^{2}-1$, $a^{11}-a^{10}-10a^{9}+9a^{8}+36a^{7}-28a^{6}-56a^{5}+35a^{4}+35a^{3}-15a^{2}-6a+1$, $a^{6}-6a^{4}+9a^{2}-2$, $a^{3}-3a$, $a^{9}-9a^{7}+27a^{5}-30a^{3}+9a$, $a^{5}-a^{4}-4a^{3}+3a^{2}+3a-1$, $a^{18}-18a^{16}+135a^{14}-546a^{12}+1287a^{10}-1782a^{8}+1386a^{6}-540a^{4}+81a^{2}-2$, $a^{19}-19a^{17}+152a^{15}-665a^{13}+1729a^{11}-2717a^{9}+2508a^{7}-1254a^{5}+285a^{3}-19a$, $a^{9}-a^{8}-8a^{7}+7a^{6}+21a^{5}-15a^{4}-20a^{3}+10a^{2}+5a-1$, $a^{6}-5a^{4}+6a^{2}-1$, $a^{3}-a^{2}-2a+1$, $a^{13}-a^{12}-12a^{11}+11a^{10}+55a^{9}-45a^{8}-120a^{7}+84a^{6}+126a^{5}-70a^{4}-56a^{3}+21a^{2}+7a-1$, $a^{4}-3a^{2}+1$, $a^{2}-a-1$, $a$, $a^{10}-a^{9}-9a^{8}+8a^{7}+28a^{6}-21a^{5}-35a^{4}+20a^{3}+15a^{2}-5a-1$, $a^{16}-15a^{14}+91a^{12}-286a^{10}+495a^{8}-462a^{6}+210a^{4}-36a^{2}+1$, $a^{3}-2a$, $a^{4}-4a^{2}+2$, $a^{14}-13a^{12}+66a^{10}-165a^{8}+210a^{6}-126a^{4}+28a^{2}-1$, $a^{8}-8a^{6}+20a^{4}-16a^{2}+2$, $a^{22}-a^{21}-21a^{20}+21a^{19}+189a^{18}-189a^{17}-951a^{16}+951a^{15}+2925a^{14}-2925a^{13}-5643a^{12}+5643a^{11}+6733a^{10}-6733a^{9}-4707a^{8}+4707a^{7}+1728a^{6}-1728a^{5}-274a^{4}+274a^{3}+12a^{2}-12a$ (assuming GRH)
Regulator:		\( 68215743661.4 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{23}\cdot(2\pi)^{0}\cdot 68215743661.4 \cdot 1}{2\cdot\sqrt{6111571184724799803076702357055363809}}\cr\approx \mathstrut & 0.115735897897 \end{aligned}\] (assuming GRH)

Galois group

$C_{23}$ (as 23T1):

A cyclic group of order 23

The 23 conjugacy class representatives for $C_{23}$

Character table for $C_{23}$ is not computed

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	$23$	$23$	$23$	$23$	$23$	$23$	$23$	$23$	$23$	$23$	$23$	$23$	$23$	$23$	R	$23$	$23$

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
\(47\)	47.23.22.1	$x^{23} + 47$	$23$	$1$	$22$	$C_{23}$	$[\ ]_{23}$