Number field 38.0.16942169463783688515274712980459055357466629107345696737681001440761438215014074919167299388239251805665492992.1

• Label: 38.0.169...992.1
• Degree: $38$
• Signature: $[0, 19]$
• Discriminant: $-1.694\times 10^{109}$
• Root discriminant: \(748.94\)
• Ramified primes: $2,19$
• Class number: not computed
• Class group: not computed
• Galois group: $C_{38}$ (as 38T1)

Normalized defining polynomial

x^38 - 304*x^36 - 266*x^35 + 41933*x^34 + 69730*x^33 - 3426479*x^32 - 8099624*x^31 + 183049762*x^30 + 548687586*x^29 - 6680231090*x^28 - 23987190756*x^27 + 169686384783*x^26 + 708938225002*x^25 - 3018350895246*x^24 - 14489790513992*x^23 + 37802280147544*x^22 + 208280775345998*x^21 - 338007964348740*x^20 - 2157118911607202*x^19 + 2151786701480777*x^18 + 16600188595793984*x^17 - 8092991352668802*x^16 - 93536887230591734*x^15 + 8981124468818326*x^14 + 410584246763815678*x^13 + 169889908746117158*x^12 - 1213081286172833338*x^11 - 877922600330127862*x^10 + 3318553597434145946*x^9 + 5125749038813794988*x^8 - 1873842236788886072*x^7 + 1268339370593460529*x^6 + 14808781213729150014*x^5 + 46758207820852858725*x^4 + 17480877413656313060*x^3 + 169357167185673333495*x^2 + 207600614196418504062*x + 391269434132253227219

Invariants

Degree:		$38$
Signature:		$[0, 19]$
Discriminant:		-16942169463783688515274712980459055357466629107345696737681001440761438215014074919167299388239251805665492992
Root discriminant:		\(748.94\)
Galois root discriminant:		$2^{3/2}19^{36/19}\approx 748.9391840287906$
Ramified primes:		\(2\), \(19\)
Discriminant root field:		\(\Q(\sqrt{-2}) \)
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:		$C_{38}$
This field is Galois and abelian over $\Q$.
Conductor:		\(2888=2^{3}\cdot 19^{2}\)
Dirichlet character group:		$\lbrace$$\chi_{2888}(2433,·)$, $\chi_{2888}(1027,·)$, $\chi_{2888}(1,·)$, $\chi_{2888}(1673,·)$, $\chi_{2888}(2699,·)$, $\chi_{2888}(115,·)$, $\chi_{2888}(913,·)$, $\chi_{2888}(1939,·)$, $\chi_{2888}(2585,·)$, $\chi_{2888}(153,·)$, $\chi_{2888}(1179,·)$, $\chi_{2888}(1825,·)$, $\chi_{2888}(2851,·)$, $\chi_{2888}(305,·)$, $\chi_{2888}(1065,·)$, $\chi_{2888}(2091,·)$, $\chi_{2888}(2737,·)$, $\chi_{2888}(1331,·)$, $\chi_{2888}(1977,·)$, $\chi_{2888}(571,·)$, $\chi_{2888}(1217,·)$, $\chi_{2888}(2243,·)$, $\chi_{2888}(457,·)$, $\chi_{2888}(1483,·)$, $\chi_{2888}(2129,·)$, $\chi_{2888}(419,·)$, $\chi_{2888}(723,·)$, $\chi_{2888}(267,·)$, $\chi_{2888}(1369,·)$, $\chi_{2888}(2395,·)$, $\chi_{2888}(609,·)$, $\chi_{2888}(1635,·)$, $\chi_{2888}(2281,·)$, $\chi_{2888}(875,·)$, $\chi_{2888}(1521,·)$, $\chi_{2888}(2547,·)$, $\chi_{2888}(761,·)$, $\chi_{2888}(1787,·)$$\rbrace$
This is a CM field.
Reflex fields:		unavailable$^{262144}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $\frac{1}{307}a^{34}+\frac{3}{307}a^{33}+\frac{90}{307}a^{32}+\frac{64}{307}a^{31}+\frac{114}{307}a^{30}+\frac{144}{307}a^{29}+\frac{15}{307}a^{28}+\frac{109}{307}a^{27}+\frac{136}{307}a^{26}+\frac{29}{307}a^{25}-\frac{147}{307}a^{24}-\frac{18}{307}a^{23}+\frac{34}{307}a^{22}+\frac{114}{307}a^{21}-\frac{115}{307}a^{20}+\frac{58}{307}a^{19}-\frac{86}{307}a^{18}-\frac{108}{307}a^{17}+\frac{73}{307}a^{16}+\frac{65}{307}a^{15}-\frac{98}{307}a^{14}+\frac{133}{307}a^{13}+\frac{42}{307}a^{12}+\frac{65}{307}a^{11}-\frac{6}{307}a^{10}+\frac{123}{307}a^{9}-\frac{151}{307}a^{8}+\frac{6}{307}a^{7}+\frac{59}{307}a^{6}+\frac{45}{307}a^{5}-\frac{93}{307}a^{4}+\frac{144}{307}a^{3}+\frac{26}{307}a^{2}-\frac{146}{307}a+\frac{102}{307}$, $\frac{1}{307}a^{35}+\frac{81}{307}a^{33}+\frac{101}{307}a^{32}-\frac{78}{307}a^{31}+\frac{109}{307}a^{30}-\frac{110}{307}a^{29}+\frac{64}{307}a^{28}+\frac{116}{307}a^{27}-\frac{72}{307}a^{26}+\frac{73}{307}a^{25}+\frac{116}{307}a^{24}+\frac{88}{307}a^{23}+\frac{12}{307}a^{22}-\frac{150}{307}a^{21}+\frac{96}{307}a^{20}+\frac{47}{307}a^{19}+\frac{150}{307}a^{18}+\frac{90}{307}a^{17}+\frac{153}{307}a^{16}+\frac{14}{307}a^{15}+\frac{120}{307}a^{14}-\frac{50}{307}a^{13}-\frac{61}{307}a^{12}+\frac{106}{307}a^{11}+\frac{141}{307}a^{10}+\frac{94}{307}a^{9}+\frac{152}{307}a^{8}+\frac{41}{307}a^{7}-\frac{132}{307}a^{6}+\frac{79}{307}a^{5}+\frac{116}{307}a^{4}-\frac{99}{307}a^{3}+\frac{83}{307}a^{2}-\frac{74}{307}a+\frac{1}{307}$, $\frac{1}{88\cdots 89}a^{36}+\frac{72\cdots 41}{88\cdots 89}a^{35}-\frac{89\cdots 42}{88\cdots 89}a^{34}+\frac{24\cdots 49}{88\cdots 89}a^{33}-\frac{24\cdots 98}{88\cdots 89}a^{32}-\frac{66\cdots 66}{88\cdots 89}a^{31}+\frac{34\cdots 56}{88\cdots 89}a^{30}+\frac{28\cdots 47}{88\cdots 89}a^{29}-\frac{22\cdots 38}{88\cdots 89}a^{28}+\frac{41\cdots 64}{88\cdots 89}a^{27}+\frac{36\cdots 93}{88\cdots 89}a^{26}-\frac{41\cdots 94}{88\cdots 89}a^{25}+\frac{44\cdots 58}{88\cdots 89}a^{24}+\frac{29\cdots 60}{88\cdots 89}a^{23}-\frac{24\cdots 44}{88\cdots 89}a^{22}+\frac{13\cdots 27}{88\cdots 89}a^{21}+\frac{10\cdots 26}{88\cdots 89}a^{20}-\frac{11\cdots 58}{88\cdots 89}a^{19}+\frac{42\cdots 92}{88\cdots 89}a^{18}-\frac{29\cdots 63}{88\cdots 89}a^{17}+\frac{54\cdots 61}{88\cdots 89}a^{16}-\frac{10\cdots 41}{88\cdots 89}a^{15}+\frac{49\cdots 77}{88\cdots 89}a^{14}+\frac{40\cdots 30}{88\cdots 89}a^{13}-\frac{49\cdots 59}{88\cdots 89}a^{12}-\frac{41\cdots 10}{88\cdots 89}a^{11}+\frac{42\cdots 09}{88\cdots 89}a^{10}-\frac{22\cdots 07}{88\cdots 89}a^{9}-\frac{33\cdots 88}{88\cdots 89}a^{8}+\frac{65\cdots 72}{88\cdots 89}a^{7}-\frac{38\cdots 66}{88\cdots 89}a^{6}+\frac{28\cdots 43}{88\cdots 89}a^{5}+\frac{16\cdots 80}{88\cdots 89}a^{4}+\frac{25\cdots 77}{88\cdots 89}a^{3}+\frac{16\cdots 72}{88\cdots 89}a^{2}+\frac{29\cdots 95}{88\cdots 89}a+\frac{17\cdots 85}{88\cdots 89}$, $\frac{1}{14\cdots 57}a^{37}+\frac{13\cdots 77}{14\cdots 57}a^{36}+\frac{17\cdots 66}{14\cdots 57}a^{35}-\frac{91\cdots 22}{14\cdots 57}a^{34}-\frac{65\cdots 51}{14\cdots 57}a^{33}+\frac{14\cdots 87}{14\cdots 57}a^{32}+\frac{45\cdots 40}{14\cdots 57}a^{31}-\frac{17\cdots 47}{14\cdots 57}a^{30}-\frac{48\cdots 81}{14\cdots 57}a^{29}+\frac{31\cdots 00}{14\cdots 57}a^{28}-\frac{17\cdots 49}{14\cdots 57}a^{27}+\frac{64\cdots 11}{14\cdots 57}a^{26}-\frac{30\cdots 94}{14\cdots 57}a^{25}+\frac{21\cdots 21}{14\cdots 57}a^{24}-\frac{67\cdots 63}{14\cdots 57}a^{23}-\frac{28\cdots 45}{14\cdots 57}a^{22}-\frac{58\cdots 87}{14\cdots 57}a^{21}+\frac{55\cdots 16}{14\cdots 57}a^{20}+\frac{25\cdots 10}{14\cdots 57}a^{19}-\frac{59\cdots 83}{14\cdots 57}a^{18}-\frac{60\cdots 81}{14\cdots 57}a^{17}+\frac{65\cdots 48}{14\cdots 57}a^{16}-\frac{52\cdots 25}{14\cdots 57}a^{15}+\frac{60\cdots 91}{14\cdots 57}a^{14}+\frac{47\cdots 47}{14\cdots 57}a^{13}+\frac{10\cdots 26}{14\cdots 57}a^{12}-\frac{19\cdots 01}{14\cdots 57}a^{11}-\frac{18\cdots 83}{14\cdots 57}a^{10}+\frac{45\cdots 59}{14\cdots 57}a^{9}+\frac{67\cdots 07}{14\cdots 57}a^{8}+\frac{38\cdots 72}{14\cdots 57}a^{7}+\frac{10\cdots 58}{36\cdots 13}a^{6}+\frac{32\cdots 42}{14\cdots 57}a^{5}-\frac{19\cdots 67}{14\cdots 57}a^{4}+\frac{69\cdots 88}{14\cdots 57}a^{3}+\frac{59\cdots 58}{14\cdots 57}a^{2}+\frac{77\cdots 46}{14\cdots 57}a-\frac{37\cdots 72}{14\cdots 57}$

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

Class group and class number

Ideal class group:		not computed
Narrow class group:		not computed
Relative class number:		data not computed

Unit group

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

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{0}\cdot(2\pi)^{19}\cdot R \cdot h}{2\cdot\sqrt{16942169463783688515274712980459055357466629107345696737681001440761438215014074919167299388239251805665492992}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

Galois group

$C_{38}$ (as 38T1):

A cyclic group of order 38

The 38 conjugacy class representatives for $C_{38}$

Character table for $C_{38}$

Intermediate fields

\(\Q(\sqrt{-2}) \), 19.19.10842505080063916320800450434338728415281531281.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	$19^{2}$	$38$	$38$	$19^{2}$	$38$	$19^{2}$	R	$38$	$38$	$38$	$38$	$19^{2}$	$19^{2}$	$38$	$38$	$19^{2}$

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
\(2\)		Deg $38$	$2$	$19$	$57$
\(19\)	19.1.19.36a18.1	$x^{19} + 342 x^{18} + 19$	$19$	$1$	$36$	$C_{19}$	$$[2]$$
	19.1.19.36a18.1	$x^{19} + 342 x^{18} + 19$	$19$	$1$	$36$	$C_{19}$	$$[2]$$

Spectrum of ring of integers