Number field 38.38.2242277458404766702002557226129081814951173635373007362690887769154380596585299665301923770160675048828125.1

• Label: 38.38.224...125.1
• Degree: $38$
• Signature: $[38, 0]$
• Discriminant: $2.242\times 10^{105}$
• Root discriminant: \(592.09\)
• Ramified primes: $5,19$
• Class number: not computed
• Class group: not computed
• Galois group: $C_{38}$ (as 38T1)

Normalized defining polynomial

x^38 - 19*x^37 - 190*x^36 + 5263*x^35 + 10963*x^34 - 655956*x^33 + 271263*x^32 + 48829164*x^31 - 74272197*x^30 - 2427590879*x^29 + 4963339201*x^28 + 85311795747*x^27 - 192448926397*x^26 - 2186062812785*x^25 + 4980640240893*x^24 + 41507384238113*x^23 - 90602195976460*x^22 - 587394556501288*x^21 + 1185165731831058*x^20 + 6181335356273740*x^19 - 11252186392848870*x^18 - 47909517088369504*x^17 + 77654230949000120*x^16 + 268901707689505353*x^15 - 387932611852231165*x^14 - 1065553844073230755*x^13 + 1387879177866434969*x^12 + 2872799715815077105*x^11 - 3477750883069386065*x^10 - 4978197957785195915*x^9 + 5844343521008310247*x^8 + 5024712299211630779*x^7 - 6066136922383939789*x^6 - 2388621891807945980*x^5 + 3307037404639155292*x^4 + 219641714027477785*x^3 - 641666806588881139*x^2 + 40067232399563857*x + 14911143898147099

Invariants

Degree:		$38$
Signature:		$[38, 0]$
Discriminant:		2242277458404766702002557226129081814951173635373007362690887769154380596585299665301923770160675048828125
Root discriminant:		\(592.09\)
Galois root discriminant:		$5^{1/2}19^{36/19}\approx 592.0884126197448$
Ramified primes:		\(5\), \(19\)
Discriminant root field:		\(\Q(\sqrt{5}) \)
$\card{ \Gal(K/\Q) }$:		$38$
This field is Galois and abelian over $\Q$.
Conductor:		\(1805=5\cdot 19^{2}\)
Dirichlet character group:		$\lbrace$$\chi_{1805}(1,·)$, $\chi_{1805}(514,·)$, $\chi_{1805}(134,·)$, $\chi_{1805}(1426,·)$, $\chi_{1805}(1046,·)$, $\chi_{1805}(1559,·)$, $\chi_{1805}(666,·)$, $\chi_{1805}(1179,·)$, $\chi_{1805}(286,·)$, $\chi_{1805}(799,·)$, $\chi_{1805}(419,·)$, $\chi_{1805}(39,·)$, $\chi_{1805}(1711,·)$, $\chi_{1805}(1331,·)$, $\chi_{1805}(951,·)$, $\chi_{1805}(1464,·)$, $\chi_{1805}(571,·)$, $\chi_{1805}(1084,·)$, $\chi_{1805}(191,·)$, $\chi_{1805}(704,·)$, $\chi_{1805}(324,·)$, $\chi_{1805}(1616,·)$, $\chi_{1805}(1236,·)$, $\chi_{1805}(1749,·)$, $\chi_{1805}(856,·)$, $\chi_{1805}(1369,·)$, $\chi_{1805}(476,·)$, $\chi_{1805}(989,·)$, $\chi_{1805}(96,·)$, $\chi_{1805}(609,·)$, $\chi_{1805}(229,·)$, $\chi_{1805}(1521,·)$, $\chi_{1805}(1141,·)$, $\chi_{1805}(1654,·)$, $\chi_{1805}(761,·)$, $\chi_{1805}(1274,·)$, $\chi_{1805}(381,·)$, $\chi_{1805}(894,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $\frac{1}{1571}a^{35}-\frac{92}{1571}a^{34}-\frac{694}{1571}a^{33}+\frac{766}{1571}a^{32}-\frac{157}{1571}a^{31}-\frac{308}{1571}a^{30}-\frac{716}{1571}a^{29}+\frac{731}{1571}a^{28}+\frac{432}{1571}a^{27}+\frac{267}{1571}a^{26}-\frac{309}{1571}a^{25}-\frac{333}{1571}a^{24}-\frac{243}{1571}a^{23}-\frac{642}{1571}a^{22}-\frac{79}{1571}a^{21}-\frac{669}{1571}a^{20}-\frac{435}{1571}a^{19}+\frac{23}{1571}a^{18}-\frac{54}{1571}a^{17}-\frac{358}{1571}a^{16}+\frac{338}{1571}a^{15}-\frac{701}{1571}a^{14}-\frac{374}{1571}a^{13}-\frac{452}{1571}a^{12}-\frac{356}{1571}a^{11}+\frac{377}{1571}a^{10}+\frac{581}{1571}a^{9}+\frac{333}{1571}a^{8}-\frac{525}{1571}a^{7}-\frac{440}{1571}a^{6}+\frac{749}{1571}a^{5}-\frac{614}{1571}a^{4}-\frac{424}{1571}a^{3}-\frac{462}{1571}a^{2}-\frac{361}{1571}a-\frac{23}{1571}$, $\frac{1}{88\!\cdots\!89}a^{36}+\frac{24\!\cdots\!38}{88\!\cdots\!89}a^{35}+\frac{27\!\cdots\!23}{88\!\cdots\!89}a^{34}+\frac{12\!\cdots\!91}{88\!\cdots\!89}a^{33}-\frac{10\!\cdots\!17}{88\!\cdots\!89}a^{32}+\frac{83\!\cdots\!90}{88\!\cdots\!89}a^{31}+\frac{54\!\cdots\!24}{88\!\cdots\!89}a^{30}+\frac{23\!\cdots\!15}{88\!\cdots\!89}a^{29}-\frac{22\!\cdots\!01}{88\!\cdots\!89}a^{28}-\frac{21\!\cdots\!32}{88\!\cdots\!89}a^{27}-\frac{26\!\cdots\!78}{88\!\cdots\!89}a^{26}+\frac{10\!\cdots\!02}{88\!\cdots\!89}a^{25}+\frac{36\!\cdots\!74}{88\!\cdots\!89}a^{24}-\frac{32\!\cdots\!83}{88\!\cdots\!89}a^{23}-\frac{26\!\cdots\!75}{88\!\cdots\!89}a^{22}-\frac{27\!\cdots\!53}{88\!\cdots\!89}a^{21}-\frac{24\!\cdots\!03}{88\!\cdots\!89}a^{20}+\frac{29\!\cdots\!43}{88\!\cdots\!89}a^{19}-\frac{27\!\cdots\!94}{88\!\cdots\!89}a^{18}+\frac{37\!\cdots\!69}{88\!\cdots\!89}a^{17}-\frac{36\!\cdots\!53}{88\!\cdots\!89}a^{16}+\frac{35\!\cdots\!12}{88\!\cdots\!89}a^{15}-\frac{36\!\cdots\!57}{88\!\cdots\!89}a^{14}+\frac{18\!\cdots\!40}{88\!\cdots\!89}a^{13}-\frac{14\!\cdots\!99}{88\!\cdots\!89}a^{12}-\frac{42\!\cdots\!67}{88\!\cdots\!89}a^{11}+\frac{33\!\cdots\!07}{88\!\cdots\!89}a^{10}-\frac{16\!\cdots\!00}{88\!\cdots\!89}a^{9}-\frac{39\!\cdots\!21}{88\!\cdots\!89}a^{8}-\frac{42\!\cdots\!30}{88\!\cdots\!89}a^{7}-\frac{27\!\cdots\!48}{88\!\cdots\!89}a^{6}-\frac{10\!\cdots\!64}{88\!\cdots\!89}a^{5}+\frac{15\!\cdots\!75}{88\!\cdots\!89}a^{4}+\frac{13\!\cdots\!53}{88\!\cdots\!89}a^{3}+\frac{17\!\cdots\!48}{88\!\cdots\!89}a^{2}+\frac{39\!\cdots\!95}{88\!\cdots\!89}a+\frac{11\!\cdots\!02}{22\!\cdots\!01}$, $\frac{1}{16\!\cdots\!19}a^{37}-\frac{22\!\cdots\!44}{16\!\cdots\!19}a^{36}+\frac{31\!\cdots\!71}{16\!\cdots\!19}a^{35}+\frac{82\!\cdots\!19}{16\!\cdots\!19}a^{34}+\frac{38\!\cdots\!90}{16\!\cdots\!19}a^{33}-\frac{18\!\cdots\!83}{16\!\cdots\!19}a^{32}-\frac{68\!\cdots\!15}{16\!\cdots\!19}a^{31}+\frac{44\!\cdots\!65}{16\!\cdots\!19}a^{30}-\frac{73\!\cdots\!25}{16\!\cdots\!19}a^{29}+\frac{10\!\cdots\!60}{16\!\cdots\!19}a^{28}+\frac{30\!\cdots\!36}{16\!\cdots\!19}a^{27}+\frac{29\!\cdots\!97}{16\!\cdots\!19}a^{26}-\frac{67\!\cdots\!21}{16\!\cdots\!19}a^{25}-\frac{27\!\cdots\!92}{16\!\cdots\!19}a^{24}-\frac{18\!\cdots\!07}{16\!\cdots\!19}a^{23}-\frac{24\!\cdots\!43}{16\!\cdots\!19}a^{22}+\frac{39\!\cdots\!31}{16\!\cdots\!19}a^{21}-\frac{68\!\cdots\!28}{16\!\cdots\!19}a^{20}-\frac{76\!\cdots\!51}{16\!\cdots\!19}a^{19}+\frac{11\!\cdots\!06}{16\!\cdots\!19}a^{18}+\frac{57\!\cdots\!90}{16\!\cdots\!19}a^{17}-\frac{71\!\cdots\!00}{16\!\cdots\!19}a^{16}-\frac{24\!\cdots\!71}{16\!\cdots\!19}a^{15}+\frac{47\!\cdots\!81}{16\!\cdots\!19}a^{14}+\frac{37\!\cdots\!05}{16\!\cdots\!19}a^{13}-\frac{10\!\cdots\!43}{16\!\cdots\!19}a^{12}+\frac{75\!\cdots\!69}{16\!\cdots\!19}a^{11}-\frac{32\!\cdots\!24}{16\!\cdots\!19}a^{10}-\frac{18\!\cdots\!58}{16\!\cdots\!19}a^{9}+\frac{70\!\cdots\!43}{16\!\cdots\!19}a^{8}+\frac{13\!\cdots\!25}{16\!\cdots\!19}a^{7}-\frac{45\!\cdots\!60}{16\!\cdots\!19}a^{6}-\frac{20\!\cdots\!52}{16\!\cdots\!19}a^{5}-\frac{43\!\cdots\!95}{16\!\cdots\!19}a^{4}+\frac{36\!\cdots\!32}{16\!\cdots\!19}a^{3}+\frac{32\!\cdots\!84}{16\!\cdots\!19}a^{2}-\frac{55\!\cdots\!78}{16\!\cdots\!19}a-\frac{19\!\cdots\!89}{42\!\cdots\!71}$

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 $C_{38}$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 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	$38$	$38$	R	$38$	$19^{2}$	$38$	$38$	R	$38$	$19^{2}$	$19^{2}$	$38$	$19^{2}$	$38$	$38$	$38$	$19^{2}$

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
\(5\)		Deg $38$	$2$	$19$	$19$
\(19\)	19.19.36.1	$x^{19} + 342 x^{18} + 19$	$19$	$1$	$36$	$C_{19}$	$[2]$
	19.19.36.1	$x^{19} + 342 x^{18} + 19$	$19$	$1$	$36$	$C_{19}$	$[2]$