Global number field 18.0.2094561035195319823494242687991631414291476262838089296768483.1

• Label: 18.0.20945610351...8483.1
• Degree: $18$
• Signature: $[0, 9]$
• Discriminant: $-\,3^{9}\cdot 13^{12}\cdot 37^{12}\cdot 97^{12}$
• Root discriminant: $2244.77$
• Ramified primes: $3, 13, 37, 97$
• Class number: $51214851072$ (GRH)
• Class group: $[2, 2, 2, 6, 18, 18, 396, 8316]$ (GRH)
• Galois group: $S_3 \times C_3$ (as 18T3)

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 416 x^{16} + 1257 x^{15} + 288337 x^{14} - 10786920 x^{13} + 157669824 x^{12} - 1816888320 x^{11} + 30380700672 x^{10} - 623446253568 x^{9} + 11510437699584 x^{8} - 113780248805376 x^{7} + 677775794503680 x^{6} - 3079806388273152 x^{5} + 18983977004040192 x^{4} - 90314630466895872 x^{3} + 453434281285386240 x^{2} - 722517043735166976 x + 436217609315155968 \)

Invariants

Degree:		$18$
Signature:		$[0, 9]$
Discriminant:		\(-2094561035195319823494242687991631414291476262838089296768483=-\,3^{9}\cdot 13^{12}\cdot 37^{12}\cdot 97^{12}\)
Root discriminant:		$2244.77$
Ramified primes:		$3, 13, 37, 97$
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{12} a^{5} - \frac{1}{12} a^{4} - \frac{1}{3} a^{3} + \frac{1}{12} a^{2} + \frac{1}{4} a$, $\frac{1}{48} a^{6} + \frac{1}{48} a^{5} + \frac{17}{48} a^{3} - \frac{19}{48} a^{2}$, $\frac{1}{576} a^{7} - \frac{1}{192} a^{6} + \frac{1}{36} a^{5} - \frac{13}{192} a^{4} - \frac{239}{576} a^{3} + \frac{11}{24} a^{2} + \frac{1}{12} a$, $\frac{1}{13824} a^{8} - \frac{1}{4608} a^{7} + \frac{1}{216} a^{6} + \frac{131}{4608} a^{5} + \frac{1681}{13824} a^{4} - \frac{187}{576} a^{3} + \frac{37}{144} a^{2} + \frac{1}{24} a - \frac{1}{3}$, $\frac{1}{55296} a^{9} + \frac{1}{55296} a^{8} - \frac{11}{13824} a^{7} - \frac{215}{55296} a^{6} - \frac{1739}{55296} a^{5} + \frac{343}{13824} a^{4} - \frac{19}{576} a^{3} - \frac{97}{288} a^{2} - \frac{1}{8} a - \frac{1}{3}$, $\frac{1}{1990656} a^{10} - \frac{5}{663552} a^{9} + \frac{1}{497664} a^{8} + \frac{355}{663552} a^{7} + \frac{17317}{1990656} a^{6} + \frac{443}{55296} a^{5} - \frac{215}{2304} a^{4} - \frac{143}{384} a^{3} - \frac{47}{288} a^{2} + \frac{1}{3} a - \frac{4}{9}$, $\frac{1}{7962624} a^{11} + \frac{1}{7962624} a^{10} + \frac{13}{1990656} a^{9} + \frac{265}{7962624} a^{8} - \frac{2507}{7962624} a^{7} + \frac{18607}{1990656} a^{6} + \frac{289}{13824} a^{5} - \frac{497}{4608} a^{4} + \frac{25}{128} a^{3} + \frac{13}{72} a^{2} - \frac{5}{18} a + \frac{2}{9}$, $\frac{1}{1146617856} a^{12} - \frac{1}{382205952} a^{11} + \frac{5}{35831808} a^{10} - \frac{2461}{382205952} a^{9} - \frac{41135}{1146617856} a^{8} + \frac{32297}{47775744} a^{7} - \frac{3905}{1492992} a^{6} + \frac{631}{55296} a^{5} - \frac{2027}{41472} a^{4} - \frac{749}{1728} a^{3} - \frac{4811}{10368} a^{2} - \frac{85}{432} a - \frac{5}{54}$, $\frac{1}{4586471424} a^{13} + \frac{1}{4586471424} a^{12} + \frac{37}{1146617856} a^{11} + \frac{169}{4586471424} a^{10} - \frac{8459}{4586471424} a^{9} + \frac{35143}{1146617856} a^{8} + \frac{22441}{47775744} a^{7} + \frac{27767}{2985984} a^{6} + \frac{1267}{165888} a^{5} + \frac{3217}{41472} a^{4} + \frac{5185}{41472} a^{3} + \frac{2995}{10368} a^{2} - \frac{23}{432} a - \frac{23}{54}$, $\frac{1}{55037657088} a^{14} - \frac{1}{18345885696} a^{13} + \frac{1}{3439853568} a^{12} - \frac{13}{18345885696} a^{11} - \frac{1967}{55037657088} a^{10} + \frac{15539}{2293235712} a^{9} + \frac{13}{1146617856} a^{8} - \frac{119}{5308416} a^{7} + \frac{751}{110592} a^{6} + \frac{589}{82944} a^{5} + \frac{9445}{497664} a^{4} + \frac{7457}{20736} a^{3} - \frac{5063}{10368} a^{2} + \frac{1}{16} a + \frac{5}{18}$, $\frac{1}{48873439494144} a^{15} - \frac{25}{16291146498048} a^{14} - \frac{1}{165112971264} a^{13} - \frac{61}{16291146498048} a^{12} - \frac{828407}{48873439494144} a^{11} - \frac{215843}{1018196656128} a^{10} - \frac{2507843}{509098328064} a^{9} + \frac{838991}{28283240448} a^{8} + \frac{4216613}{10606215168} a^{7} + \frac{1497637}{220962816} a^{6} + \frac{15002233}{441925632} a^{5} - \frac{1095737}{9206784} a^{4} - \frac{1214723}{4603392} a^{3} + \frac{96671}{255744} a^{2} - \frac{667}{2592} a - \frac{103}{3996}$, $\frac{1}{35381707959764130988032} a^{16} + \frac{29482531}{11793902653254710329344} a^{15} - \frac{49336098641}{8845426989941032747008} a^{14} - \frac{1209728376893}{11793902653254710329344} a^{13} + \frac{6027322160989}{35381707959764130988032} a^{12} - \frac{10416481174501}{327608407034853064704} a^{11} - \frac{13155321376943}{61426576319034949632} a^{10} + \frac{55847208403373}{20475525439678316544} a^{9} + \frac{135581097647935}{5118881359919579136} a^{8} + \frac{13503008800667}{213286723329982464} a^{7} - \frac{3076204982989135}{319930084994973696} a^{6} + \frac{98740223660675}{2962315601805312} a^{5} + \frac{45414637172485}{555434175338496} a^{4} + \frac{61933868477957}{185144725112832} a^{3} - \frac{391376865019}{1714303010304} a^{2} + \frac{297183423143}{642863628864} a - \frac{277663375}{1678008312}$, $\frac{1}{951885406048845275442337394292905559685028656902355361262993408} a^{17} - \frac{2235224230640057536803530255489247187133}{317295135349615091814112464764301853228342885634118453754331136} a^{16} - \frac{2084888869126891455438558505201559838404324038721}{237971351512211318860584348573226389921257164225588840315748352} a^{15} + \frac{142911439054880761700969997129136887969449638710115}{317295135349615091814112464764301853228342885634118453754331136} a^{14} - \frac{68560912728635189639579490653492509243906247417746499}{951885406048845275442337394292905559685028656902355361262993408} a^{13} - \frac{468201987093369203729029509334581961221730701273727}{26441261279134590984509372063691821102361907136176537812860928} a^{12} + \frac{175200555617640001222950849832055056614944076916605701}{3305157659891823873063671507961477637795238392022067226607616} a^{11} - \frac{64053207629313722801437167654674781943450627371799999}{550859609981970645510611917993579606299206398670344537767936} a^{10} + \frac{1047369330152503741573031584698823499519335642246218427}{137714902495492661377652979498394901574801599667586134441984} a^{9} - \frac{1116741373968048494311510623941264458836566506692205}{179316279291006069502152317055201694758856249567169445888} a^{8} + \frac{5882528837448466687529292060742561533515754363291083501}{8607181405968291336103311218649681348425099979224133402624} a^{7} + \frac{99782113307483421018456572245197928224431185272869249}{239088372388008092669536422740268926345141666089559261184} a^{6} + \frac{476085689218642517282284421736768028374681185319837245}{29886046548501011583692052842533615793142708261194907648} a^{5} - \frac{349389451502924863511726119142708645039289736095685639}{4981007758083501930615342140422269298857118043532484608} a^{4} + \frac{107850502803841782377478519522464709351775326214264385}{415083979840291827551278511701855774904759836961040384} a^{3} - \frac{7427476773275176823710342207301971784400447712339763}{25942748740018239221954906981365985931547489810065024} a^{2} - \frac{1939224235813612642090698455816855923582977570811093}{9728530777506839708233090118012244724330308678774384} a - \frac{19818929108022207704563049275560583841280018977}{69666657912311590245431884778524281203133028836}$

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{6}\times C_{18}\times C_{18}\times C_{396}\times C_{8316}$, which has order $51214851072$ (assuming GRH)

Unit group

Rank:		$8$
Torsion generator:		\( -\frac{19941217946900385909143772993538900805}{3498170993546234113386847512910045873911227314725715968} a^{17} + \frac{1613984101604668865438687146556436929}{129561888649860522718031389367038736070786196841693184} a^{16} + \frac{2094924910511702990004132350417963392705}{874542748386558528346711878227511468477806828681428992} a^{15} - \frac{2043914227268988419293043425344121196669}{388685665949581568154094168101116208212358590525079552} a^{14} - \frac{5788581565545362707875168593353753566404969}{3498170993546234113386847512910045873911227314725715968} a^{13} + \frac{17535934881009886516194546768219951657957389}{291514249462186176115570626075837156159268942893809664} a^{12} - \frac{13691449404281348378405665612837830561153401}{16195236081232565339753923670879842008848274605211648} a^{11} + \frac{2141294864961077478486554745348100986680125}{224933834461563407496582273206664472345114925072384} a^{10} - \frac{1489359612451948317956178697715921617070125373}{9109820295693318003611582064869911129977154465431552} a^{9} + \frac{95691720111535520709198734067113316798594145}{28116729307695425937072784150833059043139365634048} a^{8} - \frac{1976862183750545310502293960019944247080796063}{31631320471157354179206882169687191423531786338304} a^{7} + \frac{777522499297764379840585763128023802347459869}{1317971686298223090800286757070299642647157764096} a^{6} - \frac{1424089017304203110995650248840769310984036899}{439323895432741030266762252356766547549052588032} a^{5} + \frac{754380065178959285908594261815010599472020207}{54915486929092628783345281544595818443631573504} a^{4} - \frac{2601877667804000183997582966934812220791359585}{27457743464546314391672640772297909221815786752} a^{3} + \frac{3015258818800110688414668144209291403377860395}{6864435866136578597918160193074477305453946688} a^{2} - \frac{916613013358339860159803299947829442439046185}{429027241633536162369885012067154831590871668} a + \frac{593005038360714529435418400581890796661919}{248855708604139305318958823704846190017907} \) (order $6$)
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:		\( 723879812207986.1 \) (assuming GRH)

Galois group

$C_3\times S_3$ (as 18T3):

A solvable group of order 18

The 9 conjugacy class representatives for $S_3 \times C_3$

Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), Deg 3 x3, 3.3.1590121.2, 6.0.127947264962647622427.1, 6.0.80463854613987.1 x2, 6.0.68269089455307.2, Deg 9 x3

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 6 sibling:	data not computed
Degree 9 sibling:	data not computed

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	${\href{/LocalNumberField/2.2.0.1}{2} }^{9}$	R	${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$	${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$	${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$	R	${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$	${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$	${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$	${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$	${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$	R	${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$	${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$	${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$	${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$	${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$3$	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
$13$	13.9.6.1	$x^{9} + 234 x^{6} + 16900 x^{3} + 474552$	$3$	$3$	$6$	$C_3^2$	$[\ ]_{3}^{3}$
	13.9.6.1	$x^{9} + 234 x^{6} + 16900 x^{3} + 474552$	$3$	$3$	$6$	$C_3^2$	$[\ ]_{3}^{3}$
$37$	37.3.2.3	$x^{3} - 148$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$
	37.3.2.3	$x^{3} - 148$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$
	37.3.2.3	$x^{3} - 148$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$
	37.3.2.3	$x^{3} - 148$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$
	37.3.2.3	$x^{3} - 148$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$
	37.3.2.3	$x^{3} - 148$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$
$97$	97.9.6.1	$x^{9} + 1455 x^{6} + 696266 x^{3} + 114084125$	$3$	$3$	$6$	$C_3^2$	$[\ ]_{3}^{3}$
	97.9.6.1	$x^{9} + 1455 x^{6} + 696266 x^{3} + 114084125$	$3$	$3$	$6$	$C_3^2$	$[\ ]_{3}^{3}$