Number field 43.1.11615818670173463204840555036546350439901529918004080992211006956821939425490704861593631716403.1

• Label: 43.1.116...403.1
• Degree: $43$
• Signature: $(1, 21)$
• Discriminant: $-1.162\times 10^{94}$
• Root discriminant: \(154.01\)
• Ramified primes: see page
• Class number: not computed
• Class group: not computed
• Galois group: $S_{43}$ (as 43T10)

Normalized defining polynomial

\( x^{43} + 4x - 1 \)

Invariants

Degree:		$43$
Signature:		$(1, 21)$
Discriminant:		-11615818670173463204840555036546350439901529918004080992211006956821939425490704861593631716403
Root discriminant:		\(154.01\)
Galois root discriminant:		$181^{1/2}155556871099633^{1/2}21154104412279303^{1/2}19502370656284022684705331520651631010500317670728311348739537^{1/2}\approx 1.0777670745654398e+47$
Ramified primes:		\(181\), \(155556871099633\), \(21154104412279303\), \(19502\!\cdots\!39537\)
Discriminant root field:		$\Q(\sqrt{-11615\!\cdots\!16403}$)
$\Aut(K/\Q)$:		$C_1$
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{1}\cdot(2\pi)^{21}\cdot R \cdot h}{2\cdot\sqrt{11615818670173463204840555036546350439901529918004080992211006956821939425490704861593631716403}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

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	${\href{/padicField/2.14.0.1}{14} }^{3}{,}\,{\href{/padicField/2.1.0.1}{1} }$	$37{,}\,{\href{/padicField/3.3.0.1}{3} }{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }$	$17^{2}{,}\,{\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.1.0.1}{1} }$	$31{,}\,{\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.1.0.1}{1} }$	$43$	$40{,}\,{\href{/padicField/13.3.0.1}{3} }$	$42{,}\,{\href{/padicField/17.1.0.1}{1} }$	$35{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$	$23{,}\,18{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$	$20{,}\,{\href{/padicField/29.13.0.1}{13} }{,}\,{\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$	$30{,}\,{\href{/padicField/31.11.0.1}{11} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$	$36{,}\,{\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.1.0.1}{1} }$	$43$	${\href{/padicField/43.14.0.1}{14} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }$	$24{,}\,{\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$	$28{,}\,{\href{/padicField/53.13.0.1}{13} }{,}\,{\href{/padicField/53.2.0.1}{2} }$	$27{,}\,{\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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
\(181\)	$\Q_{181}$	$x + 179$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	181.2.1.0a1.1	$x^{2} + 177 x + 2$	$1$	$2$	$0$	$C_2$	$$[\ ]^{2}$$
	181.1.2.1a1.1	$x^{2} + 181$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
	181.2.1.0a1.1	$x^{2} + 177 x + 2$	$1$	$2$	$0$	$C_2$	$$[\ ]^{2}$$
	181.2.1.0a1.1	$x^{2} + 177 x + 2$	$1$	$2$	$0$	$C_2$	$$[\ ]^{2}$$
	181.7.1.0a1.1	$x^{7} + 4 x + 179$	$1$	$7$	$0$	$C_7$	$$[\ ]^{7}$$
		Deg $27$	$1$	$27$	$0$	$C_{27}$	$$[\ ]^{27}$$
\(155556871099633\)	$\Q_{155556871099633}$	$x$	$1$	$1$	$0$	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 $6$	$1$	$6$	$0$	$C_6$	$$[\ ]^{6}$$
		Deg $7$	$1$	$7$	$0$	$C_7$	$$[\ ]^{7}$$
		Deg $22$	$1$	$22$	$0$	22T1	$$[\ ]^{22}$$
\(21154104412279303\)	$\Q_{21154104412279303}$	$x$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	$\Q_{21154104412279303}$	$x$	$1$	$1$	$0$	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 $17$	$1$	$17$	$0$	$C_{17}$	$$[\ ]^{17}$$
		Deg $17$	$1$	$17$	$0$	$C_{17}$	$$[\ ]^{17}$$
19502370656284022684705331520651631010500317670728311348739537		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $20$	$1$	$20$	$0$	20T1	$$[\ ]^{20}$$
		Deg $21$	$1$	$21$	$0$	$C_{21}$	$$[\ ]^{21}$$

Spectrum of ring of integers