Number field 43.1.1897738148225932665352052841514539485919791655231439979873933030555398665105804881388324651.1

• Label: 43.1.189...651.1
• Degree: $43$
• Signature: $(1, 21)$
• Discriminant: $-1.898\times 10^{90}$
• Root discriminant: \(125.75\)
• Ramified primes: see page
• Class number: not computed
• Class group: not computed
• Galois group: $S_{43}$ (as 43T10)

Normalized defining polynomial

\( x^{43} - 2x - 3 \)

Invariants

Degree:		$43$
Signature:		$(1, 21)$
Discriminant:		-1897738148225932665352052841514539485919791655231439979873933030555398665105804881388324651
Root discriminant:		\(125.75\)
Galois root discriminant:		not computed
Ramified primes:		\(3\), \(139\), \(12477\!\cdots\!64601\)
Discriminant root field:		$\Q(\sqrt{-17343\!\cdots\!79539}$)
$\Aut(K/\Q)$:		$C_1$
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{1}\cdot(2\pi)^{21}\cdot R \cdot h}{2\cdot\sqrt{1897738148225932665352052841514539485919791655231439979873933030555398665105804881388324651}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

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	${\href{/padicField/2.14.0.1}{14} }^{3}{,}\,{\href{/padicField/2.1.0.1}{1} }$	R	$43$	$24{,}\,{\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.1.0.1}{1} }$	$41{,}\,{\href{/padicField/11.2.0.1}{2} }$	$42{,}\,{\href{/padicField/13.1.0.1}{1} }$	$21{,}\,{\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.1.0.1}{1} }$	$18{,}\,{\href{/padicField/19.13.0.1}{13} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.1.0.1}{1} }$	$15^{2}{,}\,{\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$	$29{,}\,{\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.6.0.1}{6} }$	$26{,}\,{\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$	$30{,}\,{\href{/padicField/37.11.0.1}{11} }{,}\,{\href{/padicField/37.2.0.1}{2} }$	$33{,}\,{\href{/padicField/41.9.0.1}{9} }{,}\,{\href{/padicField/41.1.0.1}{1} }$	${\href{/padicField/43.14.0.1}{14} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }$	$16{,}\,{\href{/padicField/47.11.0.1}{11} }{,}\,{\href{/padicField/47.9.0.1}{9} }{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.2.0.1}{2} }$	$25{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$	$25{,}\,18$

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
\(3\)	$\Q_{3}$	$x + 1$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	3.2.3.6a3.1	$x^{6} + 6 x^{5} + 18 x^{4} + 38 x^{3} + 48 x^{2} + 36 x + 11$	$3$	$2$	$6$	$C_3^2:C_4$	$$[\frac{3}{2}, \frac{3}{2}]_{2}^{2}$$
	3.6.3.18a32.1	$x^{18} + 6 x^{16} + 15 x^{14} + 6 x^{13} + 26 x^{12} + 24 x^{11} + 42 x^{10} + 42 x^{9} + 66 x^{8} + 66 x^{7} + 79 x^{6} + 78 x^{5} + 72 x^{4} + 62 x^{3} + 63 x^{2} + 42 x + 17$	$3$	$6$	$18$	18T530	not computed
	3.6.3.18a120.1	$x^{18} + 6 x^{16} + 15 x^{14} + 6 x^{13} + 26 x^{12} + 27 x^{11} + 42 x^{10} + 48 x^{9} + 66 x^{8} + 66 x^{7} + 88 x^{6} + 78 x^{5} + 78 x^{4} + 59 x^{3} + 60 x^{2} + 42 x + 23$	$3$	$6$	$18$	18T530	not computed
\(139\)	$\Q_{139}$	$x + 137$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	$\Q_{139}$	$x + 137$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	139.1.2.1a1.2	$x^{2} + 278$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
	139.2.1.0a1.1	$x^{2} + 138 x + 2$	$1$	$2$	$0$	$C_2$	$$[\ ]^{2}$$
	139.9.1.0a1.1	$x^{9} + 3 x^{3} + 70 x^{2} + 87 x + 137$	$1$	$9$	$0$	$C_9$	$$[\ ]^{9}$$
	139.10.1.0a1.1	$x^{10} + 110 x^{5} + 48 x^{4} + 130 x^{3} + 66 x^{2} + 106 x + 2$	$1$	$10$	$0$	$C_{10}$	$$[\ ]^{10}$$
	139.18.1.0a1.1	$x^{18} - x + 22$	$1$	$18$	$0$	$C_{18}$	$$[\ ]^{18}$$
124775347877419985548920353165571925958504525515910258911325684164601		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $41$	$1$	$41$	$0$	$C_{41}$	$$[\ ]^{41}$$

Spectrum of ring of integers