Number field 38.38.2596071857261356353343069145711378949501369631563408367246132297653804182620484194163230531206681625953.1

• Label: 38.38.259...953.1
• Degree: $38$
• Signature: $[38, 0]$
• Discriminant: $2.596\times 10^{102}$
• Root discriminant: \(495.58\)
• Ramified primes: $3,19$
• Class number: not computed
• Class group: not computed
• Galois group: $C_{38}$ (as 38T1)

Normalized defining polynomial

x^38 - 342*x^36 - 494*x^35 + 49571*x^34 + 131024*x^33 - 3935147*x^32 - 14319996*x^31 + 187333407*x^30 + 847596384*x^29 - 5581898660*x^28 - 30311258771*x^27 + 106432357475*x^26 + 694403262546*x^25 - 1319990424275*x^24 - 10588090616078*x^23 + 11003488266809*x^22 + 110692914430763*x^21 - 67774979212045*x^20 - 811070932087692*x^19 + 375013687116124*x^18 + 4205390929184442*x^17 - 2088691296591085*x^16 - 15265383022918127*x^15 + 9967315841901733*x^14 + 37228914395189218*x^13 - 33369608975838046*x^12 - 55529864053483792*x^11 + 69626900587988757*x^10 + 39381849534000508*x^9 - 81482325904062986*x^8 + 2939626320084510*x^7 + 44420642090413847*x^6 - 17432350286733448*x^5 - 6273911703229422*x^4 + 4327841312665199*x^3 - 44929929487917*x^2 - 220961136770617*x + 3147242965861

Invariants

Degree:		$38$
Signature:		$[38, 0]$
Discriminant:		2596071857261356353343069145711378949501369631563408367246132297653804182620484194163230531206681625953
Root discriminant:		\(495.58\)
Galois root discriminant:		$3^{1/2}19^{73/38}\approx 495.5798017530623$
Ramified primes:		\(3\), \(19\)
Discriminant root field:		\(\Q(\sqrt{57}) \)
$\card{ \Gal(K/\Q) }$:		$38$
This field is Galois and abelian over $\Q$.
Conductor:		\(1083=3\cdot 19^{2}\)
Dirichlet character group:		$\lbrace$$\chi_{1083}(512,·)$, $\chi_{1083}(1,·)$, $\chi_{1083}(514,·)$, $\chi_{1083}(1027,·)$, $\chi_{1083}(1025,·)$, $\chi_{1083}(398,·)$, $\chi_{1083}(911,·)$, $\chi_{1083}(400,·)$, $\chi_{1083}(913,·)$, $\chi_{1083}(284,·)$, $\chi_{1083}(797,·)$, $\chi_{1083}(286,·)$, $\chi_{1083}(799,·)$, $\chi_{1083}(170,·)$, $\chi_{1083}(683,·)$, $\chi_{1083}(172,·)$, $\chi_{1083}(685,·)$, $\chi_{1083}(56,·)$, $\chi_{1083}(569,·)$, $\chi_{1083}(58,·)$, $\chi_{1083}(571,·)$, $\chi_{1083}(455,·)$, $\chi_{1083}(968,·)$, $\chi_{1083}(457,·)$, $\chi_{1083}(970,·)$, $\chi_{1083}(341,·)$, $\chi_{1083}(854,·)$, $\chi_{1083}(343,·)$, $\chi_{1083}(856,·)$, $\chi_{1083}(1082,·)$, $\chi_{1083}(227,·)$, $\chi_{1083}(740,·)$, $\chi_{1083}(229,·)$, $\chi_{1083}(742,·)$, $\chi_{1083}(113,·)$, $\chi_{1083}(626,·)$, $\chi_{1083}(115,·)$, $\chi_{1083}(628,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $\frac{1}{68\!\cdots\!89}a^{37}+\frac{19\!\cdots\!15}{68\!\cdots\!89}a^{36}-\frac{28\!\cdots\!78}{68\!\cdots\!89}a^{35}+\frac{28\!\cdots\!37}{68\!\cdots\!89}a^{34}+\frac{10\!\cdots\!25}{68\!\cdots\!89}a^{33}+\frac{93\!\cdots\!71}{68\!\cdots\!89}a^{32}+\frac{56\!\cdots\!92}{23\!\cdots\!73}a^{31}+\frac{27\!\cdots\!99}{68\!\cdots\!89}a^{30}-\frac{17\!\cdots\!40}{68\!\cdots\!89}a^{29}-\frac{24\!\cdots\!38}{68\!\cdots\!89}a^{28}-\frac{57\!\cdots\!95}{68\!\cdots\!89}a^{27}+\frac{60\!\cdots\!30}{68\!\cdots\!89}a^{26}-\frac{26\!\cdots\!19}{68\!\cdots\!89}a^{25}+\frac{28\!\cdots\!34}{68\!\cdots\!89}a^{24}-\frac{86\!\cdots\!63}{68\!\cdots\!89}a^{23}+\frac{12\!\cdots\!68}{68\!\cdots\!89}a^{22}-\frac{13\!\cdots\!84}{68\!\cdots\!89}a^{21}-\frac{25\!\cdots\!66}{68\!\cdots\!89}a^{20}+\frac{28\!\cdots\!00}{68\!\cdots\!89}a^{19}+\frac{18\!\cdots\!14}{68\!\cdots\!89}a^{18}-\frac{13\!\cdots\!26}{68\!\cdots\!89}a^{17}+\frac{33\!\cdots\!71}{68\!\cdots\!89}a^{16}+\frac{20\!\cdots\!33}{68\!\cdots\!89}a^{15}+\frac{28\!\cdots\!82}{68\!\cdots\!89}a^{14}+\frac{96\!\cdots\!65}{68\!\cdots\!89}a^{13}-\frac{87\!\cdots\!34}{68\!\cdots\!89}a^{12}-\frac{26\!\cdots\!47}{68\!\cdots\!89}a^{11}-\frac{27\!\cdots\!18}{68\!\cdots\!89}a^{10}+\frac{15\!\cdots\!64}{68\!\cdots\!89}a^{9}-\frac{30\!\cdots\!25}{68\!\cdots\!89}a^{8}-\frac{20\!\cdots\!83}{68\!\cdots\!89}a^{7}-\frac{43\!\cdots\!83}{68\!\cdots\!89}a^{6}+\frac{24\!\cdots\!90}{68\!\cdots\!89}a^{5}-\frac{23\!\cdots\!32}{68\!\cdots\!89}a^{4}-\frac{10\!\cdots\!22}{68\!\cdots\!89}a^{3}-\frac{31\!\cdots\!61}{68\!\cdots\!89}a^{2}+\frac{29\!\cdots\!84}{68\!\cdots\!89}a+\frac{31\!\cdots\!65}{68\!\cdots\!89}$

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

\(\Q(\sqrt{57}) \), 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	$19^{2}$	R	$38$	$19^{2}$	$38$	$38$	$38$	R	$38$	$19^{2}$	$38$	$38$	$19^{2}$	$19^{2}$	$38$	$19^{2}$	$19^{2}$

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
\(3\)		Deg $38$	$2$	$19$	$19$
\(19\)		Deg $38$	$38$	$1$	$73$