Number field 46.2.2198573809976436140308455840261067288915834777310563784825797656919642572984790479136920131622197.1

• Label: 46.2.219...197.1
• Degree: $46$
• Signature: $[2, 22]$
• Discriminant: $2.199\times 10^{96}$
• Root discriminant: $124.28$
• Ramified primes: see page
• Class number: not computed
• Class group: not computed
• Galois group: $S_{46}$ (as 46T56)

Normalized defining polynomial

$x^{46} - 3x - 2$

Invariants

Degree:		$46$
Signature:		$[2, 22]$
Discriminant:		2198573809976436140308455840261067288915834777310563784825797656919642572984790479136920131622197
Root discriminant:		$124.28$
Galois root discriminant:		$7^{1/2}47^{1/2}4697389783^{1/2}17089161961^{1/2}2184815465401^{1/2}733058611398596401^{1/2}51977373060652904686863524314133359876448011^{1/2}\approx 1.482758850918259e+48$
Ramified primes:		$7$, $47$, $4697389783$, $17089161961$, $2184815465401$, $733058611398596401$, $51977\!\cdots\!48011$
Discriminant root field:		$\Q(\sqrt{21985\!\cdots\!22197}$)
$\Aut(K/\Q)$:		$C_1$
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 $a$)

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

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

$\begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{2}\cdot(2\pi)^{22}\cdot R \cdot h}{2\cdot\sqrt{2198573809976436140308455840261067288915834777310563784825797656919642572984790479136920131622197}}\cr\mathstrut & \text{ some values not computed } \end{aligned}$

Galois group

$S_{46}$ (as 46T56):

A non-solvable group of order 5502622159812088949850305428800254892961651752960000000000

The 105558 conjugacy class representatives for $S_{46}$ are not computed

Character table for $S_{46}$

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	${\href{/padicField/2.12.0.1}{12} }^{2}{,}\,{\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.4.0.1}{4} }^{3}{,}\,{\href{/padicField/2.2.0.1}{2} }{,}\,{\href{/padicField/2.1.0.1}{1} }^{2}$	$22^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }$	$22{,}\,18{,}\,{\href{/padicField/5.6.0.1}{6} }$	R	$17{,}\,15{,}\,{\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.2.0.1}{2} }$	$34{,}\,{\href{/padicField/13.12.0.1}{12} }$	$22{,}\,15{,}\,{\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.1.0.1}{1} }$	$32{,}\,{\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.3.0.1}{3} }$	$26{,}\,{\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.8.0.1}{8} }$	$41{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$	$35{,}\,{\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.1.0.1}{1} }$	$40{,}\,{\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.1.0.1}{1} }$	$26{,}\,{\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.8.0.1}{8} }$	${\href{/padicField/43.13.0.1}{13} }{,}\,{\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.11.0.1}{11} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$	R	$45{,}\,{\href{/padicField/53.1.0.1}{1} }$	$43{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
$7$	7.1.2.1a1.1	$x^{2} + 7$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	7.2.1.0a1.1	$x^{2} + 6 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $42$	$1$	$42$	$0$	$C_{42}$	$[\ ]^{42}$
$47$	47.1.2.1a1.1	$x^{2} + 47$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	47.2.1.0a1.1	$x^{2} + 45 x + 5$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	47.4.1.0a1.1	$x^{4} + 8 x^{2} + 40 x + 5$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
	47.4.1.0a1.1	$x^{4} + 8 x^{2} + 40 x + 5$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
	47.4.1.0a1.1	$x^{4} + 8 x^{2} + 40 x + 5$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
	47.6.1.0a1.1	$x^{6} + 2 x^{4} + 35 x^{3} + 9 x^{2} + 41 x + 5$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
	47.12.1.0a1.1	$x^{12} + 46 x^{7} + 40 x^{6} + 35 x^{5} + 12 x^{4} + 46 x^{3} + 14 x^{2} + 9 x + 5$	$1$	$12$	$0$	$C_{12}$	$[\ ]^{12}$
	47.12.1.0a1.1	$x^{12} + 46 x^{7} + 40 x^{6} + 35 x^{5} + 12 x^{4} + 46 x^{3} + 14 x^{2} + 9 x + 5$	$1$	$12$	$0$	$C_{12}$	$[\ ]^{12}$
$4697389783$	$\Q_{4697389783}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $12$	$1$	$12$	$0$	$C_{12}$	$[\ ]^{12}$
		Deg $31$	$1$	$31$	$0$	$C_{31}$	$[\ ]^{31}$
$17089161961$		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $5$	$1$	$5$	$0$	$C_5$	$[\ ]^{5}$
		Deg $11$	$1$	$11$	$0$	$C_{11}$	$[\ ]^{11}$
		Deg $12$	$1$	$12$	$0$	$C_{12}$	$[\ ]^{12}$
		Deg $16$	$1$	$16$	$0$	$C_{16}$	$[\ ]^{16}$
$2184815465401$	$\Q_{2184815465401}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{2184815465401}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
		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 $7$	$1$	$7$	$0$	$C_7$	$[\ ]^{7}$
		Deg $25$	$1$	$25$	$0$	$C_{25}$	$[\ ]^{25}$
$733058611398596401$	$\Q_{733058611398596401}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $41$	$1$	$41$	$0$	$C_{41}$	$[\ ]^{41}$
51977373060652904686863524314133359876448011	$\Q_{51\!\cdots\!11}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $8$	$1$	$8$	$0$	$C_8$	$[\ ]^{8}$
		Deg $13$	$1$	$13$	$0$	$C_{13}$	$[\ ]^{13}$
		Deg $22$	$1$	$22$	$0$	22T1	$[\ ]^{22}$

Spectrum of ring of integers