Global number field 18.0.26792203422728825738155242257180552307.1

• Label: 18.0.26792203422...2307.1
• Degree: $18$
• Signature: $[0, 9]$
• Discriminant: $-\,3^{44}\cdot 67^{9}$
• Root discriminant: $120.04$
• Ramified primes: $3, 67$
• Class number: $2206717$ (GRH)
• Class group: $[2206717]$ (GRH)
• Galois group: $C_{18}$ (as 18T1)

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 171 x^{16} - 1164 x^{15} + 12600 x^{14} - 68544 x^{13} + 539184 x^{12} - 2394342 x^{11} + 14927580 x^{10} - 54191978 x^{9} + 278675451 x^{8} - 813324834 x^{7} + 3515927145 x^{6} - 7905023649 x^{5} + 28924863876 x^{4} - 45528034884 x^{3} + 140699400039 x^{2} - 119129457897 x + 307625764301 \)

Invariants

Degree:		$18$
Signature:		$[0, 9]$
Discriminant:		\(-26792203422728825738155242257180552307=-\,3^{44}\cdot 67^{9}\)
Root discriminant:		$120.04$
Ramified primes:		$3, 67$
This field is Galois and abelian over $\Q$.
Conductor:		\(1809=3^{3}\cdot 67\)
Dirichlet character group:		$\lbrace$$\chi_{1809}(1408,·)$, $\chi_{1809}(1,·)$, $\chi_{1809}(1540,·)$, $\chi_{1809}(133,·)$, $\chi_{1809}(1609,·)$, $\chi_{1809}(202,·)$, $\chi_{1809}(1741,·)$, $\chi_{1809}(334,·)$, $\chi_{1809}(403,·)$, $\chi_{1809}(535,·)$, $\chi_{1809}(604,·)$, $\chi_{1809}(736,·)$, $\chi_{1809}(805,·)$, $\chi_{1809}(937,·)$, $\chi_{1809}(1006,·)$, $\chi_{1809}(1138,·)$, $\chi_{1809}(1207,·)$, $\chi_{1809}(1339,·)$$\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}{5310312787767278821188742063687541618943088461624910392169117206529} a^{17} - \frac{1992211276498264380272130367766224449027545771961819456774594525522}{5310312787767278821188742063687541618943088461624910392169117206529} a^{16} - \frac{2229433221163981029188600178077653828887451602927867656708014770902}{5310312787767278821188742063687541618943088461624910392169117206529} a^{15} - \frac{481261605367007036660241787975633033627000536430348773378808470703}{5310312787767278821188742063687541618943088461624910392169117206529} a^{14} - \frac{1675302502482729160922408766479400022415814376777667776484115062755}{5310312787767278821188742063687541618943088461624910392169117206529} a^{13} - \frac{1054118990561995488289670390785937145519283894075586227174648356567}{5310312787767278821188742063687541618943088461624910392169117206529} a^{12} - \frac{1365192283595859221459696060758440296328892016279137141006592991901}{5310312787767278821188742063687541618943088461624910392169117206529} a^{11} - \frac{2013383360652450966700463976891724945069962125631606429414490237225}{5310312787767278821188742063687541618943088461624910392169117206529} a^{10} + \frac{1187116687809798688413738157628762256444833349648568871017032754600}{5310312787767278821188742063687541618943088461624910392169117206529} a^{9} + \frac{2117438063830244397193716959811231133223152988677467996994854213375}{5310312787767278821188742063687541618943088461624910392169117206529} a^{8} + \frac{1615644514659211327298736907146470276297706897085836657908221957208}{5310312787767278821188742063687541618943088461624910392169117206529} a^{7} + \frac{2041761154753969367411316836895644915984651804207554872227569873184}{5310312787767278821188742063687541618943088461624910392169117206529} a^{6} - \frac{911331743538259648275314752571818767147523696234353180769200426761}{5310312787767278821188742063687541618943088461624910392169117206529} a^{5} - \frac{2371117703439577706375867965818015328558141096638477385183049701438}{5310312787767278821188742063687541618943088461624910392169117206529} a^{4} - \frac{1016904100897843364556213041955616024337834303184012301243681652541}{5310312787767278821188742063687541618943088461624910392169117206529} a^{3} - \frac{1189851379841040511885285099737516013158518714368225302367884435418}{5310312787767278821188742063687541618943088461624910392169117206529} a^{2} - \frac{423559384111360223718254248254147773131007763681569234538845790493}{5310312787767278821188742063687541618943088461624910392169117206529} a + \frac{6831832804792510515449725003623175585897536237542892884277358480}{49629091474460549730735907137266744102271854781541218618402964547}$

Class group and class number

$C_{2206717}$, which has order $2206717$ (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

\(\Q(\sqrt{-67}) \), \(\Q(\zeta_{9})^+\), 6.0.1973306043.4, \(\Q(\zeta_{27})^+\)

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	$18$	R	$18$	$18$	$18$	$18$	${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$	${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$	${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$	${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$	$18$	${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$	$18$	$18$	${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$	${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$	${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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	Data not computed
67	Data not computed