Global number field 18.0.118026364303948717624477794198045747039.1

• Label: 18.0.11802636430...7039.1
• Degree: $18$
• Signature: $[0, 9]$
• Discriminant: $-\,3^{44}\cdot 79^{9}$
• Root discriminant: $130.35$
• Ramified primes: $3, 79$
• Class number: $4980185$ (GRH)
• Class group: $[4980185]$ (GRH)
• Galois group: $C_{18}$ (as 18T1)

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 198 x^{16} - 1380 x^{15} + 17028 x^{14} - 95760 x^{13} + 850872 x^{12} - 3920490 x^{11} + 27457146 x^{10} - 103473026 x^{9} + 595838997 x^{8} - 1802014938 x^{7} + 8711912985 x^{6} - 20223847899 x^{5} + 82799864595 x^{4} - 133810000764 x^{3} + 463870664253 x^{2} - 400059814653 x + 1164793689863 \)

Invariants

Degree:		$18$
Signature:		$[0, 9]$
Discriminant:		\(-118026364303948717624477794198045747039=-\,3^{44}\cdot 79^{9}\)
Root discriminant:		$130.35$
Ramified primes:		$3, 79$
This field is Galois and abelian over $\Q$.
Conductor:		\(2133=3^{3}\cdot 79\)
Dirichlet character group:		$\lbrace$$\chi_{2133}(1,·)$, $\chi_{2133}(2053,·)$, $\chi_{2133}(712,·)$, $\chi_{2133}(394,·)$, $\chi_{2133}(1423,·)$, $\chi_{2133}(1105,·)$, $\chi_{2133}(1816,·)$, $\chi_{2133}(475,·)$, $\chi_{2133}(157,·)$, $\chi_{2133}(1186,·)$, $\chi_{2133}(868,·)$, $\chi_{2133}(1897,·)$, $\chi_{2133}(1579,·)$, $\chi_{2133}(238,·)$, $\chi_{2133}(949,·)$, $\chi_{2133}(631,·)$, $\chi_{2133}(1660,·)$, $\chi_{2133}(1342,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{1441092124932452219460866416585032360932844674687362422564884089713217} a^{17} - \frac{491062951429040340715010013617964518296288177276665316599323137442836}{1441092124932452219460866416585032360932844674687362422564884089713217} a^{16} + \frac{259176158372532195828234970413609377213540439274470098649042413894386}{1441092124932452219460866416585032360932844674687362422564884089713217} a^{15} + \frac{256662983209186302934643220989637777967013976250934573433075928041667}{1441092124932452219460866416585032360932844674687362422564884089713217} a^{14} + \frac{204818412919197127996723207037049259535214418276617619789316531148579}{1441092124932452219460866416585032360932844674687362422564884089713217} a^{13} + \frac{251836461130981502669092929873435996624054075614721745398167674832056}{1441092124932452219460866416585032360932844674687362422564884089713217} a^{12} + \frac{456966078363652203894958565793169953416810597219264967698595998092201}{1441092124932452219460866416585032360932844674687362422564884089713217} a^{11} + \frac{177665717030285358382213437860742810363916713113917980759803989577592}{1441092124932452219460866416585032360932844674687362422564884089713217} a^{10} - \frac{260822788332224548422650585473888615011887136863032472946666052963346}{1441092124932452219460866416585032360932844674687362422564884089713217} a^{9} + \frac{645957093485543038199774602280832148878594780033341552697174644046428}{1441092124932452219460866416585032360932844674687362422564884089713217} a^{8} + \frac{54603953644529028832083454964466739772093185303377541253104730232386}{1441092124932452219460866416585032360932844674687362422564884089713217} a^{7} + \frac{257461541556309051197248147907626386722344083646185674306341991849049}{1441092124932452219460866416585032360932844674687362422564884089713217} a^{6} + \frac{680160452813925970596992259676542906196347974087587301363000261708163}{1441092124932452219460866416585032360932844674687362422564884089713217} a^{5} + \frac{253764453057170408607875324173126602234785453293047767175974639033213}{1441092124932452219460866416585032360932844674687362422564884089713217} a^{4} - \frac{368672693581110772472648293126767471782806853213249505583832674780597}{1441092124932452219460866416585032360932844674687362422564884089713217} a^{3} + \frac{358663057877032194388255742162204342032226911137655988653545416772554}{1441092124932452219460866416585032360932844674687362422564884089713217} a^{2} + \frac{543486885307993879716895798393451828272789006493300494838775702581667}{1441092124932452219460866416585032360932844674687362422564884089713217} a - \frac{628128432936709037466839701810474006848947098963290928343806901302311}{1441092124932452219460866416585032360932844674687362422564884089713217}$

Class group and class number

$C_{4980185}$, which has order $4980185$ (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:		\( 40934.03294431194 \) (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

\(\Q(\sqrt{-79}) \), \(\Q(\zeta_{9})^+\), 6.0.3234828879.4, \(\Q(\zeta_{27})^+\)

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	${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$	R	${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$	$18$	${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$	${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$	${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$	${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$	${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$	$18$	${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$	${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$	$18$	$18$	$18$	${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$	$18$

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
3	Data not computed
79	Data not computed