Number field 43.1.17193642429484970947547009316647777700494014568443353175841873665979443.1

• Label: 43.1.171...443.1
• Degree: $43$
• Signature: $(1, 21)$
• Discriminant: $-1.719\times 10^{70}$
• Root discriminant: \(42.99\)
• Ramified primes: see page
• Class number: not computed
• Class group: not computed
• Galois group: $S_{43}$ (as 43T10)

Normalized defining polynomial

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

Invariants

Degree:		$43$
Signature:		$(1, 21)$
Discriminant:		\(-17193642429484970947547009316647777700494014568443353175841873665979443\) \(\medspace = -\,109\cdot 809\cdot 397519523\cdot 49\!\cdots\!61\)
Root discriminant:		\(42.99\)
Galois root discriminant:		$109^{1/2}809^{1/2}397519523^{1/2}490494801020731074863763714627948664201245545640512874061^{1/2}\approx 1.3112453023551685e+35$
Ramified primes:		\(109\), \(809\), \(397519523\), \(49049\!\cdots\!74061\)
Discriminant root field:		$\Q(\sqrt{-17193\!\cdots\!79443}$)
$\Aut(K/\Q)$:		$C_1$
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{1}\cdot(2\pi)^{21}\cdot R \cdot h}{2\cdot\sqrt{17193642429484970947547009316647777700494014568443353175841873665979443}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

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	$38{,}\,{\href{/padicField/2.5.0.1}{5} }$	$37{,}\,{\href{/padicField/3.6.0.1}{6} }$	$17^{2}{,}\,{\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.1.0.1}{1} }$	$29{,}\,{\href{/padicField/7.9.0.1}{9} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }$	${\href{/padicField/11.14.0.1}{14} }{,}\,{\href{/padicField/11.13.0.1}{13} }{,}\,{\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$	$36{,}\,{\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.1.0.1}{1} }$	$21{,}\,{\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.1.0.1}{1} }$	$26{,}\,{\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$	$41{,}\,{\href{/padicField/23.2.0.1}{2} }$	${\href{/padicField/29.13.0.1}{13} }{,}\,{\href{/padicField/29.11.0.1}{11} }{,}\,{\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$	$26{,}\,{\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$	$42{,}\,{\href{/padicField/37.1.0.1}{1} }$	$38{,}\,{\href{/padicField/41.5.0.1}{5} }$	$43$	$17{,}\,15{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.4.0.1}{4} }$	$31{,}\,{\href{/padicField/53.5.0.1}{5} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$	$42{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
\(109\)	$\Q_{109}$	$x + 103$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	109.1.2.1a1.1	$x^{2} + 109$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
	109.2.1.0a1.1	$x^{2} + 108 x + 6$	$1$	$2$	$0$	$C_2$	$$[\ ]^{2}$$
	109.4.1.0a1.1	$x^{4} + 11 x^{2} + 98 x + 6$	$1$	$4$	$0$	$C_4$	$$[\ ]^{4}$$
	109.4.1.0a1.1	$x^{4} + 11 x^{2} + 98 x + 6$	$1$	$4$	$0$	$C_4$	$$[\ ]^{4}$$
		Deg $30$	$1$	$30$	$0$	$C_{30}$	$$[\ ]^{30}$$
\(809\)	$\Q_{809}$	$x$	$1$	$1$	$0$	Trivial	$$[\ ]$$
		Deg $2$	$1$	$2$	$0$	$C_2$	$$[\ ]^{2}$$
		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $4$	$1$	$4$	$0$	$C_4$	$$[\ ]^{4}$$
		Deg $5$	$1$	$5$	$0$	$C_5$	$$[\ ]^{5}$$
		Deg $13$	$1$	$13$	$0$	$C_{13}$	$$[\ ]^{13}$$
		Deg $16$	$1$	$16$	$0$	$C_{16}$	$$[\ ]^{16}$$
\(397519523\)	$\Q_{397519523}$	$x$	$1$	$1$	$0$	Trivial	$$[\ ]$$
		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $40$	$1$	$40$	$0$	$C_{40}$	$$[\ ]^{40}$$
490494801020731074863763714627948664201245545640512874061	$\Q_{49\!\cdots\!61}$	$x$	$1$	$1$	$0$	Trivial	$$[\ ]$$
		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $11$	$1$	$11$	$0$	$C_{11}$	$$[\ ]^{11}$$
		Deg $29$	$1$	$29$	$0$	$C_{29}$	$$[\ ]^{29}$$

Spectrum of ring of integers