Number field 37.37.81381208133441979421709122744091225498491936628940230588748580298513087650630871328595025812353503688138712627681.1

• Label: 37.37.813...681.1
• Degree: $37$
• Signature: $[37, 0]$
• Discriminant: $8.138\times 10^{112}$
• Root discriminant: \(1126.25\)
• Ramified prime: $37$
• Class number: not computed
• Class group: not computed
• Galois group: $C_{37}$ (as 37T1)

Normalized defining polynomial

x^37 - 666*x^35 - 481*x^34 + 193399*x^33 + 256521*x^32 - 32436827*x^31 - 60004084*x^30 + 3503452596*x^29 + 8113398850*x^28 - 257000630926*x^27 - 704292185114*x^26 + 13150853376643*x^25 + 41268101524916*x^24 - 474337598223255*x^23 - 1672482254287228*x^22 + 12032466243732809*x^21 + 47286637172470867*x^20 - 211502629431882231*x^19 - 929675982328753625*x^18 + 2498958523266151636*x^17 + 12536142236443615902*x^16 - 18791897342800992970*x^15 - 113239929214497210129*x^14 + 80927529107744817385*x^13 + 663966210336111429023*x^12 - 146546651897009713442*x^11 - 2439258160361411529393*x^10 - 154326326264692800324*x^9 + 5386184266991634344483*x^8 + 1055793633392402952768*x^7 - 6769150214799915444440*x^6 - 1552196159553019851407*x^5 + 4355671676506075568364*x^4 + 921754431147330625307*x^3 - 1108584774969076393499*x^2 - 217498966742521653325*x + 51140551819476687829

Invariants

Degree:		$37$
Signature:		$[37, 0]$
Discriminant:		81381208133441979421709122744091225498491936628940230588748580298513087650630871328595025812353503688138712627681
Root discriminant:		\(1126.25\)
Galois root discriminant:		$37^{72/37}\approx 1126.2525502057642$
Ramified primes:		\(37\)
Discriminant root field:		\(\Q\)
$\card{ \Gal(K/\Q) }$:		$37$
This field is Galois and abelian over $\Q$.
Conductor:		\(1369=37^{2}\)
Dirichlet character group:		$\lbrace$$\chi_{1369}(1,·)$, $\chi_{1369}(260,·)$, $\chi_{1369}(519,·)$, $\chi_{1369}(778,·)$, $\chi_{1369}(1037,·)$, $\chi_{1369}(1296,·)$, $\chi_{1369}(149,·)$, $\chi_{1369}(408,·)$, $\chi_{1369}(667,·)$, $\chi_{1369}(926,·)$, $\chi_{1369}(1185,·)$, $\chi_{1369}(38,·)$, $\chi_{1369}(297,·)$, $\chi_{1369}(556,·)$, $\chi_{1369}(815,·)$, $\chi_{1369}(1074,·)$, $\chi_{1369}(1333,·)$, $\chi_{1369}(186,·)$, $\chi_{1369}(445,·)$, $\chi_{1369}(704,·)$, $\chi_{1369}(963,·)$, $\chi_{1369}(1222,·)$, $\chi_{1369}(75,·)$, $\chi_{1369}(334,·)$, $\chi_{1369}(593,·)$, $\chi_{1369}(852,·)$, $\chi_{1369}(1111,·)$, $\chi_{1369}(223,·)$, $\chi_{1369}(482,·)$, $\chi_{1369}(741,·)$, $\chi_{1369}(1000,·)$, $\chi_{1369}(1259,·)$, $\chi_{1369}(112,·)$, $\chi_{1369}(371,·)$, $\chi_{1369}(630,·)$, $\chi_{1369}(889,·)$, $\chi_{1369}(1148,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $\frac{1}{35\!\cdots\!71}a^{36}+\frac{15\!\cdots\!51}{35\!\cdots\!71}a^{35}+\frac{11\!\cdots\!81}{35\!\cdots\!71}a^{34}-\frac{11\!\cdots\!47}{35\!\cdots\!71}a^{33}-\frac{41\!\cdots\!88}{35\!\cdots\!71}a^{32}-\frac{30\!\cdots\!82}{35\!\cdots\!71}a^{31}+\frac{77\!\cdots\!48}{35\!\cdots\!71}a^{30}+\frac{12\!\cdots\!26}{35\!\cdots\!71}a^{29}-\frac{53\!\cdots\!33}{35\!\cdots\!71}a^{28}-\frac{70\!\cdots\!56}{35\!\cdots\!71}a^{27}+\frac{17\!\cdots\!37}{35\!\cdots\!71}a^{26}-\frac{15\!\cdots\!23}{35\!\cdots\!71}a^{25}-\frac{10\!\cdots\!22}{35\!\cdots\!71}a^{24}+\frac{13\!\cdots\!87}{35\!\cdots\!71}a^{23}-\frac{16\!\cdots\!42}{35\!\cdots\!71}a^{22}+\frac{10\!\cdots\!94}{35\!\cdots\!71}a^{21}-\frac{13\!\cdots\!22}{35\!\cdots\!71}a^{20}-\frac{11\!\cdots\!35}{35\!\cdots\!71}a^{19}+\frac{65\!\cdots\!92}{35\!\cdots\!71}a^{18}+\frac{35\!\cdots\!78}{35\!\cdots\!71}a^{17}-\frac{70\!\cdots\!19}{35\!\cdots\!71}a^{16}+\frac{17\!\cdots\!67}{35\!\cdots\!71}a^{15}-\frac{11\!\cdots\!91}{35\!\cdots\!71}a^{14}+\frac{67\!\cdots\!89}{35\!\cdots\!71}a^{13}+\frac{27\!\cdots\!92}{35\!\cdots\!71}a^{12}-\frac{15\!\cdots\!56}{35\!\cdots\!71}a^{11}+\frac{13\!\cdots\!77}{35\!\cdots\!71}a^{10}+\frac{11\!\cdots\!34}{35\!\cdots\!71}a^{9}+\frac{17\!\cdots\!39}{35\!\cdots\!71}a^{8}+\frac{11\!\cdots\!35}{35\!\cdots\!71}a^{7}-\frac{16\!\cdots\!49}{35\!\cdots\!71}a^{6}-\frac{17\!\cdots\!31}{35\!\cdots\!71}a^{5}+\frac{16\!\cdots\!36}{35\!\cdots\!71}a^{4}+\frac{13\!\cdots\!11}{35\!\cdots\!71}a^{3}+\frac{14\!\cdots\!47}{35\!\cdots\!71}a^{2}+\frac{13\!\cdots\!18}{35\!\cdots\!71}a-\frac{10\!\cdots\!11}{35\!\cdots\!71}$

Monogenic:		No
Index:		$1$
Inessential primes:		None

Class group and class number

not computed

Unit group

Rank:		$36$
Torsion generator:		\( -1 \)  (order $2$)
Fundamental units:		not computed
Regulator:		not computed

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{37}\cdot(2\pi)^{0}\cdot R \cdot h}{2\cdot\sqrt{81381208133441979421709122744091225498491936628940230588748580298513087650630871328595025812353503688138712627681}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

Galois group

$C_{37}$ (as 37T1):

A cyclic group of order 37

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

Character table for $C_{37}$

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	$37$	$37$	$37$	$37$	$37$	$37$	$37$	$37$	$37$	$37$	$37$	R	$37$	$37$	$37$	$37$	$37$

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
\(37\)		Deg $37$	$37$	$1$	$72$