Number field 37.1.1584269131698073346547516360976706218258582973296556645981584548317239027557.1

• Label: 37.1.158...557.1
• Degree: $37$
• Signature: $(1, 18)$
• Discriminant: $1.584\times 10^{75}$
• Root discriminant: \(107.75\)
• Ramified primes: $3,37$
• Class number: not computed
• Class group: not computed
• Galois group: $F_{37}$ (as 37T9)

Normalized defining polynomial

\( x^{37} - 3 \)

Invariants

Degree:		$37$
Signature:		$(1, 18)$
Discriminant:		\(1584269131698073346547516360976706218258582973296556645981584548317239027557\) \(\medspace = 3^{36}\cdot 37^{37}\)
Root discriminant:		\(107.75\)
Galois root discriminant:		$3^{36/37}37^{1367/1332}\approx 118.47707168667975$
Ramified primes:		\(3\), \(37\)
Discriminant root field:		\(\Q(\sqrt{37}) \)
$\Aut(K/\Q)$:		$C_1$
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 \(a\))

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

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

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{1}\cdot(2\pi)^{18}\cdot R \cdot h}{2\cdot\sqrt{1584269131698073346547516360976706218258582973296556645981584548317239027557}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

Galois group

$F_{37}$ (as 37T9):

A solvable group of order 1332

The 37 conjugacy class representatives for $F_{37}$

Character table for $F_{37}$

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	$36{,}\,{\href{/padicField/2.1.0.1}{1} }$	R	$36{,}\,{\href{/padicField/5.1.0.1}{1} }$	${\href{/padicField/7.9.0.1}{9} }^{4}{,}\,{\href{/padicField/7.1.0.1}{1} }$	${\href{/padicField/11.6.0.1}{6} }^{6}{,}\,{\href{/padicField/11.1.0.1}{1} }$	$36{,}\,{\href{/padicField/13.1.0.1}{1} }$	$36{,}\,{\href{/padicField/17.1.0.1}{1} }$	$36{,}\,{\href{/padicField/19.1.0.1}{1} }$	${\href{/padicField/23.12.0.1}{12} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }$	${\href{/padicField/29.12.0.1}{12} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }$	${\href{/padicField/31.4.0.1}{4} }^{9}{,}\,{\href{/padicField/31.1.0.1}{1} }$	R	$18^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$	${\href{/padicField/43.4.0.1}{4} }^{9}{,}\,{\href{/padicField/43.1.0.1}{1} }$	${\href{/padicField/47.3.0.1}{3} }^{12}{,}\,{\href{/padicField/47.1.0.1}{1} }$	${\href{/padicField/53.9.0.1}{9} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }$	$36{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
\(3\)		Deg $37$	$37$	$1$	$36$
\(37\)		Deg $37$	$37$	$1$	$37$

Spectrum of ring of integers