Global number field 18.0.3981163315919720752483315106924610739.1

• Label: 18.0.39811633159...0739.1
• Degree: $18$
• Signature: $[0, 9]$
• Discriminant: $-\,19^{9}\cdot 37^{16}$
• Root discriminant: $107.98$
• Ramified primes: $19, 37$
• Class number: $145669$ (GRH)
• Class group: $[145669]$ (GRH)
• Galois group: $C_{18}$ (as 18T1)

Normalized defining polynomial

\( x^{18} - 7 x^{17} + 33 x^{16} - 106 x^{15} + 513 x^{14} - 1911 x^{13} + 8905 x^{12} - 28707 x^{11} + 113286 x^{10} - 312000 x^{9} + 1072652 x^{8} - 2391737 x^{7} + 6978401 x^{6} - 11975323 x^{5} + 30556226 x^{4} - 38411315 x^{3} + 83255438 x^{2} - 59181109 x + 96478091 \)

Invariants

Degree:		$18$
Signature:		$[0, 9]$
Discriminant:		\(-3981163315919720752483315106924610739=-\,19^{9}\cdot 37^{16}\)
Root discriminant:		$107.98$
Ramified primes:		$19, 37$
This field is Galois and abelian over $\Q$.
Conductor:		\(703=19\cdot 37\)
Dirichlet character group:		$\lbrace$$\chi_{703}(1,·)$, $\chi_{703}(322,·)$, $\chi_{703}(645,·)$, $\chi_{703}(379,·)$, $\chi_{703}(75,·)$, $\chi_{703}(588,·)$, $\chi_{703}(514,·)$, $\chi_{703}(343,·)$, $\chi_{703}(417,·)$, $\chi_{703}(419,·)$, $\chi_{703}(229,·)$, $\chi_{703}(552,·)$, $\chi_{703}(493,·)$, $\chi_{703}(303,·)$, $\chi_{703}(305,·)$, $\chi_{703}(626,·)$, $\chi_{703}(248,·)$, $\chi_{703}(571,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{43} a^{15} + \frac{10}{43} a^{14} + \frac{17}{43} a^{13} + \frac{11}{43} a^{12} + \frac{13}{43} a^{11} + \frac{20}{43} a^{10} - \frac{18}{43} a^{9} + \frac{4}{43} a^{8} + \frac{5}{43} a^{7} + \frac{11}{43} a^{6} + \frac{6}{43} a^{5} + \frac{12}{43} a^{4} - \frac{11}{43} a^{2} - \frac{16}{43} a - \frac{6}{43}$, $\frac{1}{97309} a^{16} + \frac{249}{97309} a^{15} - \frac{9117}{97309} a^{14} - \frac{27144}{97309} a^{13} + \frac{43707}{97309} a^{12} + \frac{42859}{97309} a^{11} - \frac{578}{1333} a^{10} + \frac{11999}{97309} a^{9} - \frac{16239}{97309} a^{8} + \frac{44421}{97309} a^{7} + \frac{41077}{97309} a^{6} - \frac{4230}{97309} a^{5} + \frac{42385}{97309} a^{4} - \frac{46494}{97309} a^{3} - \frac{4064}{97309} a^{2} + \frac{14230}{97309} a - \frac{20311}{97309}$, $\frac{1}{354444690504687344018851216356591317119754553065543} a^{17} + \frac{943791704343810263122051173770727699019496242}{354444690504687344018851216356591317119754553065543} a^{16} + \frac{140450615627423676095506097705961034332594046409}{354444690504687344018851216356591317119754553065543} a^{15} - \frac{150630897070597802904094716650605544466033408536178}{354444690504687344018851216356591317119754553065543} a^{14} + \frac{4007776215791604951559027347993743100573114130977}{11433699693699591742543587624406171519992082356953} a^{13} - \frac{81316997165796259623859767222009199700470284012897}{354444690504687344018851216356591317119754553065543} a^{12} - \frac{86407038682579555485694413880363515621018210928776}{354444690504687344018851216356591317119754553065543} a^{11} - \frac{100797364990782654736657720664966553502041240637872}{354444690504687344018851216356591317119754553065543} a^{10} + \frac{79824043281375286661005973400389638517040641079003}{354444690504687344018851216356591317119754553065543} a^{9} + \frac{97580262985652418471161191989992612205385547925654}{354444690504687344018851216356591317119754553065543} a^{8} + \frac{465039527876614175253447450818022309683157195944}{8242899779178775442298865496664914351622198908501} a^{7} - \frac{82822069679869983744521508316165199969126364427732}{354444690504687344018851216356591317119754553065543} a^{6} - \frac{1509359495261214589119588778076319473951226590}{2378823426205955328985578633265713537716473510507} a^{5} + \frac{142184673299044187782436201960196521429370127622708}{354444690504687344018851216356591317119754553065543} a^{4} - \frac{51545453448447285215522957196082606684185829811423}{354444690504687344018851216356591317119754553065543} a^{3} + \frac{59769906879069382426518399220540684538780697172841}{354444690504687344018851216356591317119754553065543} a^{2} + \frac{94585715035713206237292522680087825151322918474327}{354444690504687344018851216356591317119754553065543} a - \frac{82243725531826559049476088499032862875809951788730}{354444690504687344018851216356591317119754553065543}$

Class group and class number

$C_{145669}$, which has order $145669$ (assuming GRH)

Unit group

Rank:		$8$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
Regulator:		\( 409151.3102125697 \) (assuming GRH)

Galois group

$C_{18}$ (as 18T1):

A cyclic group of order 18

The 18 conjugacy class representatives for $C_{18}$

Character table for $C_{18}$

Intermediate fields

\(\Q(\sqrt{-19}) \), 3.3.1369.1, 6.0.12854870299.1, 9.9.3512479453921.1

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	$18$	$18$	${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$	${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$	${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$	$18$	${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$	R	${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$	${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$	${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$	R	$18$	${\href{/LocalNumberField/43.1.0.1}{1} }^{18}$	${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$	$18$	$18$

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
19	Data not computed
37	Data not computed