Number field 37.37.6760346576864373546418554592723962033312970679630764740849754684931757688487032687926115035176212801.1

• Label: 37.37.676...801.1
• Degree: $37$
• Signature: $[37, 0]$
• Discriminant: $6.760\times 10^{99}$
• Root discriminant: \(499.01\)
• Ramified prime: $593$
• Class number: not computed
• Class group: not computed
• Galois group: $C_{37}$ (as 37T1)

Normalized defining polynomial

x^37 - x^36 - 288*x^35 + 203*x^34 + 35910*x^33 - 17336*x^32 - 2554843*x^31 + 842615*x^30 + 115283443*x^29 - 27504461*x^28 - 3475263507*x^27 + 687931612*x^26 + 71915662407*x^25 - 13974595438*x^24 - 1034496050617*x^23 + 221330910643*x^22 + 10365088714029*x^21 - 2534502436630*x^20 - 71813314605418*x^19 + 19740632343658*x^18 + 338381715805464*x^17 - 99305962871851*x^16 - 1055972317034939*x^15 + 302779333586209*x^14 + 2104618406390649*x^13 - 505910437546894*x^12 - 2560928454934750*x^11 + 376597618487442*x^10 + 1787314591059035*x^9 - 20429122626927*x^8 - 643936928000107*x^7 - 89391122404607*x^6 + 89662863870162*x^5 + 25891375086732*x^4 + 1113091773432*x^3 - 112807611860*x^2 - 8478121776*x - 123515969

Invariants

Degree:		$37$
Signature:		$[37, 0]$
Discriminant:		6760346576864373546418554592723962033312970679630764740849754684931757688487032687926115035176212801
Root discriminant:		\(499.01\)
Galois root discriminant:		$593^{36/37}\approx 499.007725403986$
Ramified primes:		\(593\)
Discriminant root field:		\(\Q\)
$\card{ \Gal(K/\Q) }$:		$37$
This field is Galois and abelian over $\Q$.
Conductor:		\(593\)
Dirichlet character group:		$\lbrace$$\chi_{593}(256,·)$, $\chi_{593}(1,·)$, $\chi_{593}(258,·)$, $\chi_{593}(399,·)$, $\chi_{593}(16,·)$, $\chi_{593}(529,·)$, $\chi_{593}(148,·)$, $\chi_{593}(277,·)$, $\chi_{593}(535,·)$, $\chi_{593}(152,·)$, $\chi_{593}(281,·)$, $\chi_{593}(154,·)$, $\chi_{593}(538,·)$, $\chi_{593}(286,·)$, $\chi_{593}(162,·)$, $\chi_{593}(220,·)$, $\chi_{593}(42,·)$, $\chi_{593}(555,·)$, $\chi_{593}(556,·)$, $\chi_{593}(306,·)$, $\chi_{593}(183,·)$, $\chi_{593}(570,·)$, $\chi_{593}(60,·)$, $\chi_{593}(62,·)$, $\chi_{593}(578,·)$, $\chi_{593}(454,·)$, $\chi_{593}(225,·)$, $\chi_{593}(311,·)$, $\chi_{593}(589,·)$, $\chi_{593}(78,·)$, $\chi_{593}(79,·)$, $\chi_{593}(345,·)$, $\chi_{593}(92,·)$, $\chi_{593}(353,·)$, $\chi_{593}(232,·)$, $\chi_{593}(367,·)$, $\chi_{593}(425,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $\frac{1}{59}a^{29}+\frac{14}{59}a^{28}+\frac{17}{59}a^{27}+\frac{6}{59}a^{26}+\frac{28}{59}a^{25}-\frac{1}{59}a^{24}+\frac{19}{59}a^{23}-\frac{10}{59}a^{22}-\frac{4}{59}a^{21}-\frac{8}{59}a^{20}-\frac{8}{59}a^{19}-\frac{3}{59}a^{18}-\frac{21}{59}a^{17}+\frac{29}{59}a^{16}-\frac{28}{59}a^{15}+\frac{11}{59}a^{14}-\frac{13}{59}a^{13}-\frac{6}{59}a^{12}-\frac{12}{59}a^{11}+\frac{16}{59}a^{10}+\frac{27}{59}a^{9}+\frac{14}{59}a^{8}-\frac{11}{59}a^{7}-\frac{13}{59}a^{6}-\frac{2}{59}a^{5}-\frac{13}{59}a^{4}+\frac{14}{59}a^{3}-\frac{17}{59}a^{2}+\frac{12}{59}a$, $\frac{1}{59}a^{30}-\frac{2}{59}a^{28}+\frac{4}{59}a^{27}+\frac{3}{59}a^{26}+\frac{20}{59}a^{25}-\frac{26}{59}a^{24}+\frac{19}{59}a^{23}+\frac{18}{59}a^{22}-\frac{11}{59}a^{21}-\frac{14}{59}a^{20}-\frac{9}{59}a^{19}+\frac{21}{59}a^{18}+\frac{28}{59}a^{17}-\frac{21}{59}a^{16}-\frac{10}{59}a^{15}+\frac{10}{59}a^{14}-\frac{1}{59}a^{13}+\frac{13}{59}a^{12}+\frac{7}{59}a^{11}-\frac{20}{59}a^{10}-\frac{10}{59}a^{9}+\frac{29}{59}a^{8}+\frac{23}{59}a^{7}+\frac{3}{59}a^{6}+\frac{15}{59}a^{5}+\frac{19}{59}a^{4}+\frac{23}{59}a^{3}+\frac{14}{59}a^{2}+\frac{9}{59}a$, $\frac{1}{59}a^{31}-\frac{27}{59}a^{28}-\frac{22}{59}a^{27}-\frac{27}{59}a^{26}-\frac{29}{59}a^{25}+\frac{17}{59}a^{24}-\frac{3}{59}a^{23}+\frac{28}{59}a^{22}-\frac{22}{59}a^{21}-\frac{25}{59}a^{20}+\frac{5}{59}a^{19}+\frac{22}{59}a^{18}-\frac{4}{59}a^{17}-\frac{11}{59}a^{16}+\frac{13}{59}a^{15}+\frac{21}{59}a^{14}-\frac{13}{59}a^{13}-\frac{5}{59}a^{12}+\frac{15}{59}a^{11}+\frac{22}{59}a^{10}+\frac{24}{59}a^{9}-\frac{8}{59}a^{8}-\frac{19}{59}a^{7}-\frac{11}{59}a^{6}+\frac{15}{59}a^{5}-\frac{3}{59}a^{4}-\frac{17}{59}a^{3}-\frac{25}{59}a^{2}+\frac{24}{59}a$, $\frac{1}{59}a^{32}+\frac{2}{59}a^{28}+\frac{19}{59}a^{27}+\frac{15}{59}a^{26}+\frac{6}{59}a^{25}+\frac{29}{59}a^{24}+\frac{10}{59}a^{23}+\frac{3}{59}a^{22}-\frac{15}{59}a^{21}+\frac{25}{59}a^{20}-\frac{17}{59}a^{19}-\frac{26}{59}a^{18}+\frac{12}{59}a^{17}+\frac{29}{59}a^{16}-\frac{27}{59}a^{15}-\frac{11}{59}a^{14}-\frac{2}{59}a^{13}-\frac{29}{59}a^{12}-\frac{7}{59}a^{11}-\frac{16}{59}a^{10}+\frac{13}{59}a^{9}+\frac{5}{59}a^{8}-\frac{13}{59}a^{7}+\frac{18}{59}a^{6}+\frac{2}{59}a^{5}-\frac{14}{59}a^{4}-\frac{1}{59}a^{3}-\frac{22}{59}a^{2}+\frac{29}{59}a$, $\frac{1}{59}a^{33}-\frac{9}{59}a^{28}-\frac{19}{59}a^{27}-\frac{6}{59}a^{26}-\frac{27}{59}a^{25}+\frac{12}{59}a^{24}+\frac{24}{59}a^{23}+\frac{5}{59}a^{22}-\frac{26}{59}a^{21}-\frac{1}{59}a^{20}-\frac{10}{59}a^{19}+\frac{18}{59}a^{18}+\frac{12}{59}a^{17}-\frac{26}{59}a^{16}-\frac{14}{59}a^{15}-\frac{24}{59}a^{14}-\frac{3}{59}a^{13}+\frac{5}{59}a^{12}+\frac{8}{59}a^{11}-\frac{19}{59}a^{10}+\frac{10}{59}a^{9}+\frac{18}{59}a^{8}-\frac{19}{59}a^{7}+\frac{28}{59}a^{6}-\frac{10}{59}a^{5}+\frac{25}{59}a^{4}+\frac{9}{59}a^{3}+\frac{4}{59}a^{2}-\frac{24}{59}a$, $\frac{1}{59}a^{34}-\frac{11}{59}a^{28}+\frac{29}{59}a^{27}+\frac{27}{59}a^{26}+\frac{28}{59}a^{25}+\frac{15}{59}a^{24}-\frac{1}{59}a^{23}+\frac{2}{59}a^{22}+\frac{22}{59}a^{21}-\frac{23}{59}a^{20}+\frac{5}{59}a^{19}-\frac{15}{59}a^{18}+\frac{21}{59}a^{17}+\frac{11}{59}a^{16}+\frac{19}{59}a^{15}-\frac{22}{59}a^{14}+\frac{6}{59}a^{13}+\frac{13}{59}a^{12}-\frac{9}{59}a^{11}-\frac{23}{59}a^{10}+\frac{25}{59}a^{9}-\frac{11}{59}a^{8}-\frac{12}{59}a^{7}-\frac{9}{59}a^{6}+\frac{7}{59}a^{5}+\frac{10}{59}a^{4}+\frac{12}{59}a^{3}-\frac{10}{59}a$, $\frac{1}{59}a^{35}+\frac{6}{59}a^{28}-\frac{22}{59}a^{27}-\frac{24}{59}a^{26}+\frac{28}{59}a^{25}-\frac{12}{59}a^{24}-\frac{25}{59}a^{23}-\frac{29}{59}a^{22}-\frac{8}{59}a^{21}-\frac{24}{59}a^{20}+\frac{15}{59}a^{19}-\frac{12}{59}a^{18}+\frac{16}{59}a^{17}-\frac{16}{59}a^{16}+\frac{24}{59}a^{15}+\frac{9}{59}a^{14}-\frac{12}{59}a^{13}-\frac{16}{59}a^{12}+\frac{22}{59}a^{11}+\frac{24}{59}a^{10}-\frac{9}{59}a^{9}+\frac{24}{59}a^{8}-\frac{12}{59}a^{7}-\frac{18}{59}a^{6}-\frac{12}{59}a^{5}-\frac{13}{59}a^{4}-\frac{23}{59}a^{3}-\frac{20}{59}a^{2}+\frac{14}{59}a$, $\frac{1}{12\!\cdots\!03}a^{36}+\frac{49\!\cdots\!95}{12\!\cdots\!03}a^{35}+\frac{24\!\cdots\!38}{12\!\cdots\!03}a^{34}-\frac{90\!\cdots\!33}{12\!\cdots\!03}a^{33}+\frac{53\!\cdots\!76}{12\!\cdots\!03}a^{32}+\frac{36\!\cdots\!38}{12\!\cdots\!03}a^{31}-\frac{60\!\cdots\!07}{12\!\cdots\!03}a^{30}-\frac{59\!\cdots\!98}{12\!\cdots\!03}a^{29}-\frac{51\!\cdots\!27}{12\!\cdots\!03}a^{28}+\frac{17\!\cdots\!50}{12\!\cdots\!03}a^{27}-\frac{44\!\cdots\!20}{12\!\cdots\!03}a^{26}-\frac{26\!\cdots\!42}{12\!\cdots\!03}a^{25}+\frac{60\!\cdots\!00}{12\!\cdots\!03}a^{24}-\frac{17\!\cdots\!93}{12\!\cdots\!03}a^{23}+\frac{31\!\cdots\!52}{12\!\cdots\!03}a^{22}+\frac{47\!\cdots\!17}{12\!\cdots\!03}a^{21}-\frac{39\!\cdots\!00}{12\!\cdots\!03}a^{20}+\frac{69\!\cdots\!35}{20\!\cdots\!17}a^{19}+\frac{10\!\cdots\!30}{12\!\cdots\!03}a^{18}+\frac{19\!\cdots\!81}{12\!\cdots\!03}a^{17}-\frac{42\!\cdots\!13}{12\!\cdots\!03}a^{16}-\frac{20\!\cdots\!67}{12\!\cdots\!03}a^{15}+\frac{13\!\cdots\!59}{12\!\cdots\!03}a^{14}-\frac{78\!\cdots\!26}{12\!\cdots\!03}a^{13}-\frac{38\!\cdots\!94}{12\!\cdots\!03}a^{12}+\frac{47\!\cdots\!62}{12\!\cdots\!03}a^{11}-\frac{42\!\cdots\!25}{12\!\cdots\!03}a^{10}-\frac{39\!\cdots\!27}{12\!\cdots\!03}a^{9}+\frac{46\!\cdots\!85}{12\!\cdots\!03}a^{8}+\frac{31\!\cdots\!14}{12\!\cdots\!03}a^{7}+\frac{46\!\cdots\!62}{12\!\cdots\!03}a^{6}+\frac{31\!\cdots\!04}{12\!\cdots\!03}a^{5}+\frac{35\!\cdots\!41}{12\!\cdots\!03}a^{4}+\frac{54\!\cdots\!19}{12\!\cdots\!03}a^{3}+\frac{60\!\cdots\!89}{12\!\cdots\!03}a^{2}-\frac{58\!\cdots\!72}{12\!\cdots\!03}a-\frac{81\!\cdots\!64}{20\!\cdots\!17}$

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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{37}\cdot(2\pi)^{0}\cdot R \cdot h}{2\cdot\sqrt{6760346576864373546418554592723962033312970679630764740849754684931757688487032687926115035176212801}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

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	$37$	$37$	$37$	$37$	$37$	$37$	$37$	$37$	$37$	$37$	$37$	$37$	$37$	$37$	$37$	$37$	${\href{/padicField/59.1.0.1}{1} }^{37}$

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
\(593\)		Deg $37$	$37$	$1$	$36$