Number field 43.1.335477317645632815204821422239306587903706814127360732280247849979340897599838242672095300419584.1

• Label: 43.1.335...584.1
• Degree: $43$
• Signature: $(1, 21)$
• Discriminant: $-3.355\times 10^{95}$
• Root discriminant: \(166.54\)
• Ramified primes: see page
• Class number: not computed
• Class group: not computed
• Galois group: $S_{43}$ (as 43T10)

Normalized defining polynomial

\( x^{43} - 3x - 4 \)

Invariants

Degree:		$43$
Signature:		$(1, 21)$
Discriminant:		-335477317645632815204821422239306587903706814127360732280247849979340897599838242672095300419584
Root discriminant:		\(166.54\)
Galois root discriminant:		not computed
Ramified primes:		\(2\), \(5736251\), \(13297\!\cdots\!32871\)
Discriminant root field:		$\Q(\sqrt{-76278\!\cdots\!06621}$)
$\Aut(K/\Q)$:		$C_1$
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 \(a\))

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

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:		$21$
Torsion generator:		\( -1 \)  (order $2$)
Fundamental units:		not computed
Regulator:		not computed
Unit signature rank:		not computed

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{1}\cdot(2\pi)^{21}\cdot R \cdot h}{2\cdot\sqrt{335477317645632815204821422239306587903706814127360732280247849979340897599838242672095300419584}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

Galois group

$S_{43}$ (as 43T10):

A non-solvable group of order 60415263063373835637355132068513997507264512000000000

The 63261 conjugacy class representatives for $S_{43}$ are not computed

Character table for $S_{43}$

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	R	$42{,}\,{\href{/padicField/3.1.0.1}{1} }$	$43$	$31{,}\,{\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.1.0.1}{1} }$	${\href{/padicField/11.14.0.1}{14} }{,}\,{\href{/padicField/11.13.0.1}{13} }{,}\,{\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$	$22{,}\,15{,}\,{\href{/padicField/13.6.0.1}{6} }$	$20{,}\,{\href{/padicField/17.11.0.1}{11} }{,}\,{\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$	$22{,}\,{\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$	$20{,}\,19{,}\,{\href{/padicField/23.4.0.1}{4} }$	$27{,}\,{\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}$	$43$	$42{,}\,{\href{/padicField/37.1.0.1}{1} }$	$21{,}\,{\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$	$42{,}\,{\href{/padicField/43.1.0.1}{1} }$	$20{,}\,{\href{/padicField/47.7.0.1}{7} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$	$35{,}\,{\href{/padicField/53.8.0.1}{8} }$	$42{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
\(2\)	$\Q_{2}$	$x + 1$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	2.1.2.2a1.1	$x^{2} + 2 x + 2$	$2$	$1$	$2$	$C_2$	$$[2]$$
	2.2.2.4a1.2	$x^{4} + 2 x^{3} + 5 x^{2} + 8 x + 5$	$2$	$2$	$4$	$C_4$	$$[2]^{2}$$
	2.3.2.6a1.2	$x^{6} + 2 x^{4} + 4 x^{3} + x^{2} + 4 x + 9$	$2$	$3$	$6$	$C_6$	$$[2]^{3}$$
	2.3.2.6a1.1	$x^{6} + 2 x^{4} + 4 x^{3} + x^{2} + 4 x + 5$	$2$	$3$	$6$	$C_6$	$$[2]^{3}$$
	2.6.2.12a1.2	$x^{12} + 2 x^{10} + 2 x^{9} + x^{8} + 4 x^{7} + 5 x^{6} + 2 x^{5} + 6 x^{4} + 8 x^{3} + x^{2} + 4 x + 5$	$2$	$6$	$12$	$C_{12}$	$$[2]^{6}$$
	2.6.2.12a1.1	$x^{12} + 2 x^{10} + 2 x^{9} + x^{8} + 4 x^{7} + 5 x^{6} + 2 x^{5} + 6 x^{4} + 4 x^{3} + x^{2} + 4 x + 5$	$2$	$6$	$12$	$C_6\times C_2$	$$[2]^{6}$$
\(5736251\)	$\Q_{5736251}$	$x$	$1$	$1$	$0$	Trivial	$$[\ ]$$
		Deg $2$	$1$	$2$	$0$	$C_2$	$$[\ ]^{2}$$
		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $38$	$1$	$38$	$0$	$C_{38}$	$$[\ ]^{38}$$
13297659175117020385300439521039825046372945434356686517822982914329253832871	$\Q_{13\!\cdots\!71}$	$x$	$1$	$1$	$0$	Trivial	$$[\ ]$$
		Deg $2$	$1$	$2$	$0$	$C_2$	$$[\ ]^{2}$$
		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $7$	$1$	$7$	$0$	$C_7$	$$[\ ]^{7}$$
		Deg $8$	$1$	$8$	$0$	$C_8$	$$[\ ]^{8}$$
		Deg $23$	$1$	$23$	$0$	$C_{23}$	$$[\ ]^{23}$$

Spectrum of ring of integers