Number field 43.1.3772838381179050124528639896195900262054652258977602756862847748815326440308126620948314666748046875.1

• Label: 43.1.377...875.1
• Degree: $43$
• Signature: $(1, 21)$
• Discriminant: $-3.773\times 10^{99}$
• Root discriminant: \(206.89\)
• Ramified primes: see page
• Class number: not computed
• Class group: not computed
• Galois group: $S_{43}$ (as 43T10)

Normalized defining polynomial

\( x^{43} - 5x - 5 \)

Invariants

Degree:		$43$
Signature:		$(1, 21)$
Discriminant:		-3772838381179050124528639896195900262054652258977602756862847748815326440308126620948314666748046875
Root discriminant:		\(206.89\)
Galois root discriminant:		$5^{42/43}1201^{1/2}3109^{1/2}627374521^{1/2}7083328981731612358415823717337673911296516278980882783^{1/2}\approx 6.2041015477474564e+35$
Ramified primes:		\(5\), \(1201\), \(3109\), \(627374521\), \(70833\!\cdots\!82783\)
Discriminant root field:		$\Q(\sqrt{-16593\!\cdots\!15187}$)
$\Aut(K/\Q)$:		$C_1$
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{1}\cdot(2\pi)^{21}\cdot R \cdot h}{2\cdot\sqrt{3772838381179050124528639896195900262054652258977602756862847748815326440308126620948314666748046875}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

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	$38{,}\,{\href{/padicField/2.5.0.1}{5} }$	$37{,}\,{\href{/padicField/3.3.0.1}{3} }{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }$	R	$20{,}\,{\href{/padicField/7.11.0.1}{11} }{,}\,{\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$	$31{,}\,{\href{/padicField/11.12.0.1}{12} }$	$21{,}\,19{,}\,{\href{/padicField/13.3.0.1}{3} }$	$20{,}\,{\href{/padicField/17.7.0.1}{7} }^{2}{,}\,{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }$	$36{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }$	$25{,}\,{\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }$	$15{,}\,{\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$	$26{,}\,15{,}\,{\href{/padicField/31.2.0.1}{2} }$	$20{,}\,15{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$	$23{,}\,{\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.1.0.1}{1} }$	$42{,}\,{\href{/padicField/43.1.0.1}{1} }$	$30{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }$	$35{,}\,{\href{/padicField/53.4.0.1}{4} }^{2}$	$17{,}\,{\href{/padicField/59.9.0.1}{9} }^{2}{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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
\(5\)		Deg $43$	$43$	$1$	$42$
\(1201\)	$\Q_{1201}$	$x$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	$\Q_{1201}$	$x$	$1$	$1$	$0$	Trivial	$$[\ ]$$
		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $39$	$1$	$39$	$0$	$C_{39}$	$$[\ ]^{39}$$
\(3109\)	$\Q_{3109}$	$x$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	$\Q_{3109}$	$x$	$1$	$1$	$0$	Trivial	$$[\ ]$$
		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $4$	$1$	$4$	$0$	$C_4$	$$[\ ]^{4}$$
		Deg $12$	$1$	$12$	$0$	$C_{12}$	$$[\ ]^{12}$$
		Deg $23$	$1$	$23$	$0$	$C_{23}$	$$[\ ]^{23}$$
\(627374521\)		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $41$	$1$	$41$	$0$	$C_{41}$	$$[\ ]^{41}$$
7083328981731612358415823717337673911296516278980882783	$\Q_{70\!\cdots\!83}$	$x$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	$\Q_{70\!\cdots\!83}$	$x$	$1$	$1$	$0$	Trivial	$$[\ ]$$
		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $5$	$1$	$5$	$0$	$C_5$	$$[\ ]^{5}$$
		Deg $6$	$1$	$6$	$0$	$C_6$	$$[\ ]^{6}$$
		Deg $7$	$1$	$7$	$0$	$C_7$	$$[\ ]^{7}$$
		Deg $8$	$1$	$8$	$0$	$C_8$	$$[\ ]^{8}$$
		Deg $13$	$1$	$13$	$0$	$C_{13}$	$$[\ ]^{13}$$

Spectrum of ring of integers