Number field 37.1.12461303884336350249094912131778654446280662104983736977381814821407046034784256.1

• Label: 37.1.124...256.1
• Degree: $37$
• Signature: $[1, 18]$
• Discriminant: $1.246\times 10^{79}$
• Root discriminant: \(137.31\)
• Ramified primes: see page
• Class number: not computed
• Class group: not computed
• Galois group: $S_{37}$ (as 37T11)

Normalized defining polynomial

\( x^{37} + 2x - 4 \)

Invariants

Degree:		$37$
Signature:		$[1, 18]$
Discriminant:		\(12461303884336350249094912131778654446280662104983736977381814821407046034784256\) \(\medspace = 2^{70}\cdot 13\cdot 137\cdot 195791\cdot 10489284688357\cdot 5723750340436657\cdot 504174169295944373011\)
Root discriminant:		\(137.31\)
Galois root discriminant:		not computed
Ramified primes:		\(2\), \(13\), \(137\), \(195791\), \(10489284688357\), \(5723750340436657\), \(504174169295944373011\)
Discriminant root field:		$\Q(\sqrt{10555\!\cdots\!02069}$)
$\Aut(K/\Q)$:		$C_1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $\frac{1}{2}a^{36}$

Monogenic:		Not computed
Index:		$1$
Inessential primes:		None

Class group and class number

Ideal class group:		not computed
Narrow class group:		not computed

Unit group

Rank:		$18$
Torsion generator:		\( -1 \)  (order $2$)
Fundamental units:		not computed
Regulator:		not computed

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{1}\cdot(2\pi)^{18}\cdot R \cdot h}{2\cdot\sqrt{12461303884336350249094912131778654446280662104983736977381814821407046034784256}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

Galois group

$S_{37}$ (as 37T11):

A non-solvable group of order 13763753091226345046315979581580902400000000

The 21637 conjugacy class representatives for $S_{37}$ are not computed

Character table for $S_{37}$

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	R	$29{,}\,{\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.3.0.1}{3} }$	$19{,}\,17{,}\,{\href{/padicField/5.1.0.1}{1} }$	$18{,}\,{\href{/padicField/7.13.0.1}{13} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$	${\href{/padicField/11.14.0.1}{14} }^{2}{,}\,{\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.1.0.1}{1} }$	R	$27{,}\,{\href{/padicField/17.10.0.1}{10} }$	$21{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.1.0.1}{1} }$	$22{,}\,15$	$27{,}\,{\href{/padicField/29.10.0.1}{10} }$	$21{,}\,15{,}\,{\href{/padicField/31.1.0.1}{1} }$	$36{,}\,{\href{/padicField/37.1.0.1}{1} }$	$28{,}\,{\href{/padicField/41.9.0.1}{9} }$	$26{,}\,{\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.1.0.1}{1} }$	$19{,}\,{\href{/padicField/47.13.0.1}{13} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$	$19{,}\,{\href{/padicField/53.11.0.1}{11} }{,}\,{\href{/padicField/53.7.0.1}{7} }$	$31{,}\,{\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
\(2\)	$\Q_{2}$	$x + 1$	$1$	$1$	$0$	Trivial	$$[\ ]$$
		Deg $36$	$36$	$1$	$70$
\(13\)	13.1.2.1a1.2	$x^{2} + 26$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
	13.2.1.0a1.1	$x^{2} + 12 x + 2$	$1$	$2$	$0$	$C_2$	$$[\ ]^{2}$$
	13.8.1.0a1.1	$x^{8} + 8 x^{4} + 12 x^{3} + 2 x^{2} + 3 x + 2$	$1$	$8$	$0$	$C_8$	$$[\ ]^{8}$$
	13.9.1.0a1.1	$x^{9} + 12 x^{4} + 8 x^{3} + 12 x^{2} + 12 x + 11$	$1$	$9$	$0$	$C_9$	$$[\ ]^{9}$$
	13.16.1.0a1.1	$x^{16} + 3 x^{8} + 12 x^{7} + 8 x^{6} + 2 x^{5} + 12 x^{4} + 9 x^{3} + 12 x^{2} + 6 x + 2$	$1$	$16$	$0$	$C_{16}$	$$[\ ]^{16}$$
\(137\)	$\Q_{137}$	$x + 134$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	137.1.2.1a1.2	$x^{2} + 411$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
	137.11.1.0a1.1	$x^{11} + x + 134$	$1$	$11$	$0$	$C_{11}$	$$[\ ]^{11}$$
	137.23.1.0a1.1	$x^{23} + 12 x + 134$	$1$	$23$	$0$	$C_{23}$	$$[\ ]^{23}$$
\(195791\)	$\Q_{195791}$	$x$	$1$	$1$	$0$	Trivial	$$[\ ]$$
		Deg $2$	$1$	$2$	$0$	$C_2$	$$[\ ]^{2}$$
		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $5$	$1$	$5$	$0$	$C_5$	$$[\ ]^{5}$$
		Deg $27$	$1$	$27$	$0$	$C_{27}$	$$[\ ]^{27}$$
\(10489284688357\)		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $5$	$1$	$5$	$0$	$C_5$	$$[\ ]^{5}$$
		Deg $30$	$1$	$30$	$0$	$C_{30}$	$$[\ ]^{30}$$
\(5723750340436657\)		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $35$	$1$	$35$	$0$	$C_{35}$	$$[\ ]^{35}$$
504174169295944373011		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $35$	$1$	$35$	$0$	$C_{35}$	$$[\ ]^{35}$$

Spectrum of ring of integers