Number field 38.38.16942169463783688515274712980459055357466629107345696737681001440761438215014074919167299388239251805665492992.1

• Label: 38.38.169...992.1
• Degree: $38$
• Signature: $[38, 0]$
• 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 - 380*x^36 - 266*x^35 + 63821*x^34 + 83562*x^33 - 6256643*x^32 - 11509288*x^31 + 398509458*x^30 + 918656194*x^29 - 17380907210*x^28 - 47344848052*x^27 + 533314140719*x^26 + 1662473883930*x^25 - 11647648756634*x^24 - 40923179473256*x^23 + 181005604480888*x^22 + 715727405895534*x^21 - 1976012600481636*x^20 - 8914335341014098*x^19 + 14710496794203337*x^18 + 78472602229296616*x^17 - 70364895826642358*x^16 - 479863268077305974*x^15 + 189419547579075974*x^14 + 1983828013159641414*x^13 - 176595262606149922*x^12 - 5351384399064377994*x^11 - 251894410670678806*x^10 + 9047670021382744322*x^9 + 314377081540580004*x^8 - 9072513198006007128*x^7 + 741757919696912561*x^6 + 4733302683037712926*x^5 - 1085560138598806567*x^4 - 932207154547761420*x^3 + 351031993292846127*x^2 - 17577545764044434*x + 235154409794327

Invariants

Degree:		$38$
Signature:		$[38, 0]$
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:		\(\Q(\sqrt{2}) \)
$\card{ \Gal(K/\Q) }$:		$38$
This field is Galois and abelian over $\Q$.
Conductor:		\(2888=2^{3}\cdot 19^{2}\)
Dirichlet character group:		$\lbrace$$\chi_{2888}(1,·)$, $\chi_{2888}(2053,·)$, $\chi_{2888}(2433,·)$, $\chi_{2888}(1673,·)$, $\chi_{2888}(1293,·)$, $\chi_{2888}(913,·)$, $\chi_{2888}(533,·)$, $\chi_{2888}(2585,·)$, $\chi_{2888}(153,·)$, $\chi_{2888}(2205,·)$, $\chi_{2888}(1825,·)$, $\chi_{2888}(1445,·)$, $\chi_{2888}(305,·)$, $\chi_{2888}(1065,·)$, $\chi_{2888}(685,·)$, $\chi_{2888}(2737,·)$, $\chi_{2888}(2357,·)$, $\chi_{2888}(1977,·)$, $\chi_{2888}(1597,·)$, $\chi_{2888}(1217,·)$, $\chi_{2888}(837,·)$, $\chi_{2888}(457,·)$, $\chi_{2888}(77,·)$, $\chi_{2888}(2509,·)$, $\chi_{2888}(2129,·)$, $\chi_{2888}(1749,·)$, $\chi_{2888}(1369,·)$, $\chi_{2888}(989,·)$, $\chi_{2888}(229,·)$, $\chi_{2888}(609,·)$, $\chi_{2888}(2661,·)$, $\chi_{2888}(2281,·)$, $\chi_{2888}(1901,·)$, $\chi_{2888}(381,·)$, $\chi_{2888}(1521,·)$, $\chi_{2888}(1141,·)$, $\chi_{2888}(761,·)$, $\chi_{2888}(2813,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $\frac{1}{11\!\cdots\!03}a^{36}-\frac{23\!\cdots\!53}{11\!\cdots\!03}a^{35}+\frac{49\!\cdots\!09}{11\!\cdots\!03}a^{34}-\frac{29\!\cdots\!21}{11\!\cdots\!03}a^{33}-\frac{31\!\cdots\!37}{11\!\cdots\!03}a^{32}+\frac{29\!\cdots\!53}{11\!\cdots\!03}a^{31}+\frac{34\!\cdots\!10}{11\!\cdots\!03}a^{30}+\frac{26\!\cdots\!16}{11\!\cdots\!03}a^{29}+\frac{50\!\cdots\!08}{11\!\cdots\!03}a^{28}+\frac{42\!\cdots\!90}{11\!\cdots\!03}a^{27}-\frac{35\!\cdots\!75}{11\!\cdots\!03}a^{26}+\frac{40\!\cdots\!48}{11\!\cdots\!03}a^{25}-\frac{10\!\cdots\!22}{11\!\cdots\!03}a^{24}-\frac{21\!\cdots\!61}{11\!\cdots\!03}a^{23}-\frac{10\!\cdots\!43}{11\!\cdots\!03}a^{22}-\frac{34\!\cdots\!75}{11\!\cdots\!03}a^{21}+\frac{21\!\cdots\!90}{11\!\cdots\!03}a^{20}+\frac{31\!\cdots\!10}{11\!\cdots\!03}a^{19}+\frac{29\!\cdots\!92}{11\!\cdots\!03}a^{18}-\frac{28\!\cdots\!77}{11\!\cdots\!03}a^{17}-\frac{81\!\cdots\!09}{11\!\cdots\!03}a^{16}-\frac{19\!\cdots\!41}{11\!\cdots\!03}a^{15}-\frac{84\!\cdots\!54}{11\!\cdots\!03}a^{14}-\frac{21\!\cdots\!96}{11\!\cdots\!03}a^{13}-\frac{97\!\cdots\!87}{88\!\cdots\!89}a^{12}-\frac{13\!\cdots\!88}{11\!\cdots\!03}a^{11}+\frac{43\!\cdots\!31}{11\!\cdots\!03}a^{10}-\frac{15\!\cdots\!31}{11\!\cdots\!03}a^{9}-\frac{36\!\cdots\!31}{11\!\cdots\!03}a^{8}-\frac{16\!\cdots\!22}{11\!\cdots\!03}a^{7}-\frac{36\!\cdots\!73}{11\!\cdots\!03}a^{6}+\frac{28\!\cdots\!08}{11\!\cdots\!03}a^{5}+\frac{53\!\cdots\!57}{11\!\cdots\!03}a^{4}-\frac{38\!\cdots\!74}{11\!\cdots\!03}a^{3}-\frac{30\!\cdots\!78}{11\!\cdots\!03}a^{2}-\frac{16\!\cdots\!64}{11\!\cdots\!03}a-\frac{45\!\cdots\!74}{11\!\cdots\!03}$, $\frac{1}{19\!\cdots\!13}a^{37}-\frac{52\!\cdots\!50}{19\!\cdots\!13}a^{36}-\frac{35\!\cdots\!86}{19\!\cdots\!13}a^{35}+\frac{81\!\cdots\!81}{19\!\cdots\!13}a^{34}+\frac{57\!\cdots\!48}{19\!\cdots\!13}a^{33}-\frac{45\!\cdots\!98}{19\!\cdots\!13}a^{32}-\frac{88\!\cdots\!88}{19\!\cdots\!13}a^{31}+\frac{10\!\cdots\!45}{19\!\cdots\!13}a^{30}+\frac{17\!\cdots\!13}{19\!\cdots\!13}a^{29}-\frac{40\!\cdots\!83}{19\!\cdots\!13}a^{28}-\frac{47\!\cdots\!62}{19\!\cdots\!13}a^{27}+\frac{28\!\cdots\!50}{19\!\cdots\!13}a^{26}-\frac{58\!\cdots\!04}{19\!\cdots\!13}a^{25}+\frac{21\!\cdots\!19}{19\!\cdots\!13}a^{24}+\frac{73\!\cdots\!69}{19\!\cdots\!13}a^{23}-\frac{83\!\cdots\!97}{19\!\cdots\!13}a^{22}+\frac{79\!\cdots\!64}{19\!\cdots\!13}a^{21}-\frac{80\!\cdots\!08}{19\!\cdots\!13}a^{20}-\frac{50\!\cdots\!65}{19\!\cdots\!13}a^{19}+\frac{51\!\cdots\!86}{19\!\cdots\!13}a^{18}-\frac{78\!\cdots\!37}{19\!\cdots\!13}a^{17}-\frac{69\!\cdots\!85}{19\!\cdots\!13}a^{16}+\frac{83\!\cdots\!65}{19\!\cdots\!13}a^{15}-\frac{29\!\cdots\!51}{19\!\cdots\!13}a^{14}-\frac{30\!\cdots\!61}{19\!\cdots\!13}a^{13}-\frac{17\!\cdots\!47}{19\!\cdots\!13}a^{12}+\frac{70\!\cdots\!15}{19\!\cdots\!13}a^{11}-\frac{93\!\cdots\!20}{19\!\cdots\!13}a^{10}-\frac{67\!\cdots\!79}{19\!\cdots\!13}a^{9}-\frac{30\!\cdots\!76}{19\!\cdots\!13}a^{8}+\frac{38\!\cdots\!69}{19\!\cdots\!13}a^{7}-\frac{84\!\cdots\!73}{19\!\cdots\!13}a^{6}-\frac{90\!\cdots\!19}{19\!\cdots\!13}a^{5}+\frac{80\!\cdots\!06}{19\!\cdots\!13}a^{4}+\frac{84\!\cdots\!23}{19\!\cdots\!13}a^{3}-\frac{54\!\cdots\!49}{19\!\cdots\!13}a^{2}+\frac{44\!\cdots\!66}{19\!\cdots\!13}a+\frac{92\!\cdots\!28}{19\!\cdots\!13}$

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

Class group and class number

not computed

Unit group

Rank:		$37$
Torsion generator:		\( -1 \)  (order $2$)
Fundamental units:		not computed
Regulator:		not computed

Class number formula

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

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	$38$	$38$	$19^{2}$	$38$	$38$	$19^{2}$	R	$19^{2}$	$38$	$19^{2}$	$38$	$19^{2}$	$38$	$19^{2}$	$38$	$38$

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 $38$	$2$	$19$	$57$
\(19\)		Deg $38$	$19$	$2$	$72$