Number field 43.1.77599287638405671151517937900886220067123731112054237412765860523695113073894359040.1

• Label: 43.1.775...040.1
• Degree: $43$
• Signature: $(1, 21)$
• Discriminant: $-7.760\times 10^{82}$
• Root discriminant: \(84.66\)
• Ramified primes: see page
• Class number: not computed
• Class group: not computed
• Galois group: $S_{43}$ (as 43T10)

Normalized defining polynomial

\( x^{43} + 2x - 2 \)

Invariants

Degree:		$43$
Signature:		$(1, 21)$
Discriminant:		-77599287638405671151517937900886220067123731112054237412765860523695113073894359040
Root discriminant:		\(84.66\)
Galois root discriminant:		$2^{42/43}5^{1/2}19^{1/2}4987^{1/2}11154877^{1/2}914818621983827673311^{1/2}3649515064680458585719692338597379997^{1/2}\approx 2.6141364721574758e+35$
Ramified primes:		\(2\), \(5\), \(19\), \(4987\), \(11154877\), \(914818621983827673311\), \(36495\!\cdots\!79997\)
Discriminant root field:		$\Q(\sqrt{-17644\!\cdots\!27635}$)
$\Aut(K/\Q)$:		$C_1$
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{1}\cdot(2\pi)^{21}\cdot R \cdot h}{2\cdot\sqrt{77599287638405671151517937900886220067123731112054237412765860523695113073894359040}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

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	$37{,}\,{\href{/padicField/3.6.0.1}{6} }$	R	$20{,}\,{\href{/padicField/7.11.0.1}{11} }{,}\,{\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$	$31{,}\,{\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.4.0.1}{4} }$	$42{,}\,{\href{/padicField/13.1.0.1}{1} }$	$40{,}\,{\href{/padicField/17.3.0.1}{3} }$	R	$29{,}\,{\href{/padicField/23.14.0.1}{14} }$	$29{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$	$30{,}\,{\href{/padicField/31.11.0.1}{11} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$	$43$	$34{,}\,{\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.1.0.1}{1} }$	${\href{/padicField/43.7.0.1}{7} }^{6}{,}\,{\href{/padicField/43.1.0.1}{1} }$	$34{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$	$20{,}\,{\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$	$27{,}\,{\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
\(2\)		Deg $43$	$43$	$1$	$42$
\(5\)	$\Q_{5}$	$x + 3$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	5.1.2.1a1.2	$x^{2} + 10$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
	5.3.1.0a1.1	$x^{3} + 3 x + 3$	$1$	$3$	$0$	$C_3$	$$[\ ]^{3}$$
	5.4.1.0a1.1	$x^{4} + 4 x^{2} + 4 x + 2$	$1$	$4$	$0$	$C_4$	$$[\ ]^{4}$$
	5.5.1.0a1.1	$x^{5} + 4 x + 3$	$1$	$5$	$0$	$C_5$	$$[\ ]^{5}$$
		Deg $28$	$1$	$28$	$0$	$C_{28}$	$$[\ ]^{28}$$
\(19\)	$\Q_{19}$	$x + 17$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	19.1.2.1a1.2	$x^{2} + 38$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
	19.5.1.0a1.1	$x^{5} + 5 x + 17$	$1$	$5$	$0$	$C_5$	$$[\ ]^{5}$$
	19.7.1.0a1.1	$x^{7} + 6 x + 17$	$1$	$7$	$0$	$C_7$	$$[\ ]^{7}$$
		Deg $28$	$1$	$28$	$0$	$C_{28}$	$$[\ ]^{28}$$
\(4987\)	$\Q_{4987}$	$x$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	$\Q_{4987}$	$x$	$1$	$1$	$0$	Trivial	$$[\ ]$$
		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $6$	$1$	$6$	$0$	$C_6$	$$[\ ]^{6}$$
		Deg $33$	$1$	$33$	$0$	$C_{33}$	$$[\ ]^{33}$$
\(11154877\)		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $6$	$1$	$6$	$0$	$C_6$	$$[\ ]^{6}$$
		Deg $35$	$1$	$35$	$0$	$C_{35}$	$$[\ ]^{35}$$
914818621983827673311	$\Q_{91\!\cdots\!11}$	$x$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	$\Q_{91\!\cdots\!11}$	$x$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	$\Q_{91\!\cdots\!11}$	$x$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	$\Q_{91\!\cdots\!11}$	$x$	$1$	$1$	$0$	Trivial	$$[\ ]$$
		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $3$	$1$	$3$	$0$	$C_3$	$$[\ ]^{3}$$
		Deg $4$	$1$	$4$	$0$	$C_4$	$$[\ ]^{4}$$
		Deg $30$	$1$	$30$	$0$	$C_{30}$	$$[\ ]^{30}$$
3649515064680458585719692338597379997		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $6$	$1$	$6$	$0$	$C_6$	$$[\ ]^{6}$$
		Deg $35$	$1$	$35$	$0$	$C_{35}$	$$[\ ]^{35}$$

Spectrum of ring of integers