Number field 47.1.3847243917163654348435987752506690356998556730699127064362242968132131231881167.1

• Label: 47.1.384...167.1
• Degree: $47$
• Signature: $[1, 23]$
• Discriminant: $-3.847\times 10^{78}$
• Root discriminant: \(46.99\)
• Ramified primes: see page
• Class number: not computed
• Class group: not computed
• Galois group: $S_{47}$ (as 47T6)

Normalized defining polynomial

\( x^{47} - x - 1 \)

Invariants

Degree:		$47$
Signature:		$[1, 23]$
Discriminant:		\(-3847243917163654348435987752506690356998556730699127064362242968132131231881167\) \(\medspace = -\,11\cdot 199\cdot 227153\cdot 393351713\cdot 3297807724117\cdot 73745876125109\cdot 80\!\cdots\!59\)
Root discriminant:		\(46.99\)
Galois root discriminant:		$11^{1/2}199^{1/2}227153^{1/2}393351713^{1/2}3297807724117^{1/2}73745876125109^{1/2}80880041787257476320797180928990259^{1/2}\approx 1.961439246360604e+39$
Ramified primes:		\(11\), \(199\), \(227153\), \(393351713\), \(3297807724117\), \(73745876125109\), \(80880\!\cdots\!90259\)
Discriminant root field:		$\Q(\sqrt{-38472\!\cdots\!81167}$)
$\Aut(K/\Q)$:		$C_1$
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $a^{42}$, $a^{43}$, $a^{44}$, $a^{45}$, $a^{46}$

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

Class group and class number

Ideal class group:		not computed
Narrow class group:		not computed

Unit group

Rank:		$23$
Torsion generator:		\( -1 \)  (order $2$)
Fundamental units:		not computed
Regulator:		not computed

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{1}\cdot(2\pi)^{23}\cdot R \cdot h}{2\cdot\sqrt{3847243917163654348435987752506690356998556730699127064362242968132131231881167}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

Galois group

$S_{47}$ (as 47T6):

A non-solvable group of order 258623241511168180642964355153611979969197632389120000000000

The 124754 conjugacy class representatives for $S_{47}$ are not computed

Character table for $S_{47}$

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	$42{,}\,{\href{/padicField/2.3.0.1}{3} }{,}\,{\href{/padicField/2.2.0.1}{2} }$	$22{,}\,16{,}\,{\href{/padicField/3.7.0.1}{7} }{,}\,{\href{/padicField/3.2.0.1}{2} }$	$46{,}\,{\href{/padicField/5.1.0.1}{1} }$	$41{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }$	R	$37{,}\,{\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.3.0.1}{3} }$	$21{,}\,18{,}\,{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }$	$28{,}\,{\href{/padicField/19.11.0.1}{11} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$	$23{,}\,17{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$	$45{,}\,{\href{/padicField/29.2.0.1}{2} }$	$28{,}\,{\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }$	$38{,}\,{\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$	$23{,}\,{\href{/padicField/41.8.0.1}{8} }^{2}{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }$	$43{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$	$47$	$35{,}\,{\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }$	$28{,}\,{\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$

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
\(11\)	11.1.2.1a1.2	$x^{2} + 22$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
	11.4.1.0a1.1	$x^{4} + 8 x^{2} + 10 x + 2$	$1$	$4$	$0$	$C_4$	$$[\ ]^{4}$$
	11.16.1.0a1.1	$x^{16} + x^{8} + 10 x^{7} + x^{6} + 3 x^{5} + 5 x^{4} + 3 x^{3} + 10 x^{2} + 9 x + 2$	$1$	$16$	$0$	$C_{16}$	$$[\ ]^{16}$$
		Deg $25$	$1$	$25$	$0$	$C_{25}$	$$[\ ]^{25}$$
\(199\)	199.1.2.1a1.1	$x^{2} + 199$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
	199.2.1.0a1.1	$x^{2} + 193 x + 3$	$1$	$2$	$0$	$C_2$	$$[\ ]^{2}$$
	199.5.1.0a1.1	$x^{5} + 3 x + 196$	$1$	$5$	$0$	$C_5$	$$[\ ]^{5}$$
	199.9.1.0a1.1	$x^{9} + 8 x^{3} + 177 x^{2} + 141 x + 196$	$1$	$9$	$0$	$C_9$	$$[\ ]^{9}$$
	199.12.1.0a1.1	$x^{12} + 33 x^{7} + 192 x^{6} + 197 x^{5} + 138 x^{4} + 69 x^{3} + 57 x^{2} + 151 x + 3$	$1$	$12$	$0$	$C_{12}$	$$[\ ]^{12}$$
	199.17.1.0a1.1	$x^{17} + 13 x + 196$	$1$	$17$	$0$	$C_{17}$	$$[\ ]^{17}$$
\(227153\)		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $3$	$1$	$3$	$0$	$C_3$	$$[\ ]^{3}$$
		Deg $42$	$1$	$42$	$0$	$C_{42}$	$$[\ ]^{42}$$
\(393351713\)		Deg $2$	$1$	$2$	$0$	$C_2$	$$[\ ]^{2}$$
		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $3$	$1$	$3$	$0$	$C_3$	$$[\ ]^{3}$$
		Deg $7$	$1$	$7$	$0$	$C_7$	$$[\ ]^{7}$$
		Deg $13$	$1$	$13$	$0$	$C_{13}$	$$[\ ]^{13}$$
		Deg $20$	$1$	$20$	$0$	20T1	$$[\ ]^{20}$$
\(3297807724117\)	$\Q_{3297807724117}$	$x$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	$\Q_{3297807724117}$	$x$	$1$	$1$	$0$	Trivial	$$[\ ]$$
		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $4$	$1$	$4$	$0$	$C_4$	$$[\ ]^{4}$$
		Deg $39$	$1$	$39$	$0$	$C_{39}$	$$[\ ]^{39}$$
\(73745876125109\)	$\Q_{73745876125109}$	$x$	$1$	$1$	$0$	Trivial	$$[\ ]$$
		Deg $2$	$1$	$2$	$0$	$C_2$	$$[\ ]^{2}$$
		Deg $2$	$1$	$2$	$0$	$C_2$	$$[\ ]^{2}$$
		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $18$	$1$	$18$	$0$	$C_{18}$	$$[\ ]^{18}$$
		Deg $22$	$1$	$22$	$0$	22T1	$$[\ ]^{22}$$
80880041787257476320797180928990259	$\Q_{80\!\cdots\!59}$	$x$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	$\Q_{80\!\cdots\!59}$	$x$	$1$	$1$	$0$	Trivial	$$[\ ]$$
		Deg $2$	$1$	$2$	$0$	$C_2$	$$[\ ]^{2}$$
		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $3$	$1$	$3$	$0$	$C_3$	$$[\ ]^{3}$$
		Deg $5$	$1$	$5$	$0$	$C_5$	$$[\ ]^{5}$$
		Deg $33$	$1$	$33$	$0$	$C_{33}$	$$[\ ]^{33}$$

Spectrum of ring of integers