Number field 43.1.17493904304575564092260553260976286616122110509580830730313660732011571.1

• Label: 43.1.174...571.1
• Degree: $43$
• Signature: $(1, 21)$
• Discriminant: $-1.749\times 10^{70}$
• Root discriminant: \(43.01\)
• 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:		\(-17493904304575564092260553260976286616122110509580830730313660732011571\) \(\medspace = -\,59\cdot 397\cdot 877\cdot 935899\cdot 2402649604770175349\cdot 37\!\cdots\!51\)
Root discriminant:		\(43.01\)
Galois root discriminant:		$59^{1/2}397^{1/2}877^{1/2}935899^{1/2}2402649604770175349^{1/2}378726079619435938610415929241631681351^{1/2}\approx 1.3226452398347625e+35$
Ramified primes:		\(59\), \(397\), \(877\), \(935899\), \(2402649604770175349\), \(37872\!\cdots\!81351\)
Discriminant root field:		$\Q(\sqrt{-17493\!\cdots\!11571}$)
$\Aut(K/\Q)$:		$C_1$
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{1}\cdot(2\pi)^{21}\cdot R \cdot h}{2\cdot\sqrt{17493904304575564092260553260976286616122110509580830730313660732011571}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

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	$38{,}\,{\href{/padicField/2.5.0.1}{5} }$	$37{,}\,{\href{/padicField/3.3.0.1}{3} }{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }$	$18{,}\,{\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.7.0.1}{7} }{,}\,{\href{/padicField/5.5.0.1}{5} }{,}\,{\href{/padicField/5.3.0.1}{3} }$	$27{,}\,{\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.1.0.1}{1} }$	$22{,}\,17{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$	$29{,}\,{\href{/padicField/13.14.0.1}{14} }$	$34{,}\,{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$	$43$	$15{,}\,{\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.2.0.1}{2} }$	$34{,}\,{\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.2.0.1}{2} }$	$42{,}\,{\href{/padicField/31.1.0.1}{1} }$	$19{,}\,{\href{/padicField/37.14.0.1}{14} }{,}\,{\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.3.0.1}{3} }$	$21{,}\,{\href{/padicField/41.13.0.1}{13} }{,}\,{\href{/padicField/41.9.0.1}{9} }$	${\href{/padicField/43.2.0.1}{2} }^{21}{,}\,{\href{/padicField/43.1.0.1}{1} }$	$20{,}\,{\href{/padicField/47.9.0.1}{9} }{,}\,{\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$	$34{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$	R

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
\(59\)	$\Q_{59}$	$x + 57$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	59.1.2.1a1.1	$x^{2} + 59$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
	59.2.1.0a1.1	$x^{2} + 58 x + 2$	$1$	$2$	$0$	$C_2$	$$[\ ]^{2}$$
	59.5.1.0a1.1	$x^{5} + 8 x + 57$	$1$	$5$	$0$	$C_5$	$$[\ ]^{5}$$
	59.5.1.0a1.1	$x^{5} + 8 x + 57$	$1$	$5$	$0$	$C_5$	$$[\ ]^{5}$$
	59.6.1.0a1.1	$x^{6} + 2 x^{4} + 18 x^{3} + 38 x^{2} + 2$	$1$	$6$	$0$	$C_6$	$$[\ ]^{6}$$
	59.7.1.0a1.1	$x^{7} + 10 x + 57$	$1$	$7$	$0$	$C_7$	$$[\ ]^{7}$$
	59.15.1.0a1.1	$x^{15} + 57 x^{6} + 24 x^{5} + 23 x^{4} + 13 x^{3} + 39 x^{2} + 58 x + 57$	$1$	$15$	$0$	$C_{15}$	$$[\ ]^{15}$$
\(397\)	$\Q_{397}$	$x$	$1$	$1$	$0$	Trivial	$$[\ ]$$
		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $40$	$1$	$40$	$0$	$C_{40}$	$$[\ ]^{40}$$
\(877\)		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $5$	$1$	$5$	$0$	$C_5$	$$[\ ]^{5}$$
		Deg $12$	$1$	$12$	$0$	$C_{12}$	$$[\ ]^{12}$$
		Deg $24$	$1$	$24$	$0$	$C_{24}$	$$[\ ]^{24}$$
\(935899\)	$\Q_{935899}$	$x$	$1$	$1$	$0$	Trivial	$$[\ ]$$
		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 $6$	$1$	$6$	$0$	$C_6$	$$[\ ]^{6}$$
		Deg $11$	$1$	$11$	$0$	$C_{11}$	$$[\ ]^{11}$$
		Deg $14$	$1$	$14$	$0$	$C_{14}$	$$[\ ]^{14}$$
\(2402649604770175349\)	$\Q_{2402649604770175349}$	$x$	$1$	$1$	$0$	Trivial	$$[\ ]$$
		Deg $2$	$1$	$2$	$0$	$C_2$	$$[\ ]^{2}$$
		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $15$	$1$	$15$	$0$	$C_{15}$	$$[\ ]^{15}$$
		Deg $23$	$1$	$23$	$0$	$C_{23}$	$$[\ ]^{23}$$
378726079619435938610415929241631681351	$\Q_{37\!\cdots\!51}$	$x$	$1$	$1$	$0$	Trivial	$$[\ ]$$
		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $40$	$1$	$40$	$0$	$C_{40}$	$$[\ ]^{40}$$

Spectrum of ring of integers