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Properties

Label 18.0.33286958328...1875.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{45}\cdot 5^{8}\cdot 19^{16}$
Root discriminant $436.64$
Ramified primes $3, 5, 19$
Class number $243$ (GRH)
Class group $[3, 9, 9]$ (GRH)
Galois group $S_3\times D_9$ (as 18T50)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![699473507584, 1686750632064, 1408298336064, 431541779712, -48798420480, -100211012352, -29706477732, 738088326, 3278237751, 687285632, 47606400, -8094114, -538365, -56772, 26163, 0, 342, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 342*x^16 + 26163*x^14 - 56772*x^13 - 538365*x^12 - 8094114*x^11 + 47606400*x^10 + 687285632*x^9 + 3278237751*x^8 + 738088326*x^7 - 29706477732*x^6 - 100211012352*x^5 - 48798420480*x^4 + 431541779712*x^3 + 1408298336064*x^2 + 1686750632064*x + 699473507584)
 
gp: K = bnfinit(x^18 + 342*x^16 + 26163*x^14 - 56772*x^13 - 538365*x^12 - 8094114*x^11 + 47606400*x^10 + 687285632*x^9 + 3278237751*x^8 + 738088326*x^7 - 29706477732*x^6 - 100211012352*x^5 - 48798420480*x^4 + 431541779712*x^3 + 1408298336064*x^2 + 1686750632064*x + 699473507584, 1)
 

Normalized defining polynomial

\( x^{18} + 342 x^{16} + 26163 x^{14} - 56772 x^{13} - 538365 x^{12} - 8094114 x^{11} + 47606400 x^{10} + 687285632 x^{9} + 3278237751 x^{8} + 738088326 x^{7} - 29706477732 x^{6} - 100211012352 x^{5} - 48798420480 x^{4} + 431541779712 x^{3} + 1408298336064 x^{2} + 1686750632064 x + 699473507584 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-332869583280588772090547978346399892639407421875=-\,3^{45}\cdot 5^{8}\cdot 19^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $436.64$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $3, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{76} a^{9} + \frac{1}{4} a^{3}$, $\frac{1}{304} a^{10} - \frac{1}{304} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} + \frac{1}{8} a^{6} + \frac{1}{8} a^{5} + \frac{1}{16} a^{4} - \frac{7}{16} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{608} a^{11} - \frac{1}{608} a^{10} - \frac{1}{304} a^{9} - \frac{1}{8} a^{8} + \frac{1}{16} a^{7} - \frac{3}{16} a^{6} + \frac{1}{32} a^{5} - \frac{7}{32} a^{4} - \frac{3}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{1216} a^{12} - \frac{1}{1216} a^{11} - \frac{1}{608} a^{10} - \frac{1}{304} a^{9} + \frac{1}{32} a^{8} + \frac{5}{32} a^{7} - \frac{15}{64} a^{6} + \frac{9}{64} a^{5} + \frac{1}{16} a^{4} - \frac{1}{2} a^{3} - \frac{3}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{6080} a^{13} - \frac{1}{1216} a^{11} + \frac{1}{760} a^{10} - \frac{3}{3040} a^{9} - \frac{1}{80} a^{8} + \frac{39}{320} a^{7} + \frac{31}{160} a^{6} + \frac{7}{64} a^{5} - \frac{33}{160} a^{4} + \frac{19}{80} a^{3} - \frac{13}{40} a^{2} - \frac{9}{20} a + \frac{1}{10}$, $\frac{1}{12160} a^{14} - \frac{1}{2432} a^{12} + \frac{1}{1520} a^{11} - \frac{3}{6080} a^{10} - \frac{1}{160} a^{9} + \frac{39}{640} a^{8} + \frac{31}{320} a^{7} - \frac{25}{128} a^{6} - \frac{33}{320} a^{5} + \frac{19}{160} a^{4} - \frac{33}{80} a^{3} + \frac{11}{40} a^{2} - \frac{9}{20} a$, $\frac{1}{170240} a^{15} - \frac{1}{170240} a^{14} + \frac{1}{34048} a^{13} - \frac{67}{170240} a^{12} - \frac{1}{2660} a^{11} + \frac{17}{17024} a^{10} + \frac{37}{170240} a^{9} - \frac{337}{8960} a^{8} + \frac{843}{8960} a^{7} + \frac{2199}{8960} a^{6} + \frac{123}{2240} a^{5} + \frac{27}{224} a^{4} - \frac{11}{56} a^{3} - \frac{17}{280} a^{2} - \frac{12}{35} a - \frac{11}{28}$, $\frac{1}{851200} a^{16} + \frac{1}{851200} a^{15} - \frac{11}{851200} a^{14} + \frac{11}{170240} a^{13} + \frac{3}{212800} a^{12} - \frac{3}{12160} a^{11} - \frac{1163}{851200} a^{10} - \frac{1457}{851200} a^{9} + \frac{563}{8960} a^{8} - \frac{37}{1280} a^{7} + \frac{71}{11200} a^{6} + \frac{1231}{5600} a^{5} + \frac{1163}{5600} a^{4} - \frac{71}{175} a^{3} + \frac{117}{700} a^{2} + \frac{11}{35} a + \frac{74}{175}$, $\frac{1}{104028522593734595122898166960461726303367573780549656501601816462226272055616000} a^{17} + \frac{10740846628831214431190531354167555346942838399217667724567108931567047559}{52014261296867297561449083480230863151683786890274828250800908231113136027808000} a^{16} + \frac{8053251983918883060617869092479335193438410966735525134563349574103129201}{6501782662108412195181135435028857893960473361284353531350113528889142003476000} a^{15} + \frac{140462932616578830768904030874944598906555285281891129237580229188528481117}{7430608756695328223064154782890123307383398127182118321542986890159019432544000} a^{14} + \frac{859304355219121534605659824918601003895772448316134487622164002550561893771}{14861217513390656446128309565780246614766796254364236643085973780318038865088000} a^{13} + \frac{103670605154333245692567292592530645626942613556867915506408143682368303967}{604816991824038343737780040467800734321904498724125909893033816640850418928000} a^{12} - \frac{1570090482105214839439162613968034107130446843868982963354942590582101609019}{14861217513390656446128309565780246614766796254364236643085973780318038865088000} a^{11} - \frac{11383772358196320993694657275820291687294477298004657620202683332900775779577}{26007130648433648780724541740115431575841893445137414125400454115556568013904000} a^{10} - \frac{17634239791016267035957538809606300250988975188319947400218370071082391237063}{2737592699835120924286793867380571744825462467909201486884258327953322948832000} a^{9} - \frac{25627707810799984238165350412472756867164714360156502539381362845660227120359}{547518539967024184857358773476114348965092493581840297376851665590664589766400} a^{8} + \frac{1331672579337204718397213877234288400912471773821049717120020395178850701922459}{5475185399670241848573587734761143489650924935818402973768516655906645897664000} a^{7} - \frac{662444765549700790247484975986219830942446624782037393546262560480876302372267}{2737592699835120924286793867380571744825462467909201486884258327953322948832000} a^{6} + \frac{31595533176372660715289672747960605158411971252421492538184037827955900941519}{136879634991756046214339693369028587241273123395460074344212916397666147441600} a^{5} + \frac{6370579086192006715325579785952157952253321853775601824766686592426766397049}{42774885934923764441981154177821433512897851061081273232566536374270671075500} a^{4} + \frac{580428179829561412073520309463088712150580568491288470682962508392650083317}{42774885934923764441981154177821433512897851061081273232566536374270671075500} a^{3} + \frac{16106138420083998427629143474426490716607444460221957874447488745809136792369}{85549771869847528883962308355642867025795702122162546465133072748541342151000} a^{2} - \frac{10372425918858907231684808594144521943951925987900631565123153602905676903279}{85549771869847528883962308355642867025795702122162546465133072748541342151000} a + \frac{6175299693728945025966946057515516207526946183565449523830132915534276585491}{42774885934923764441981154177821433512897851061081273232566536374270671075500}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{3}\times C_{9}\times C_{9}$, which has order $243$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{155380707720764876214755728214890293}{14184248916925967681624020808280581634070208000} a^{17} - \frac{606104612124657192512607875864265711}{67375182355398346487714098839332762761833488000} a^{16} + \frac{126496339720266422027125327834151369709}{33687591177699173243857049419666381380916744000} a^{15} - \frac{417263286057654698220710234098449178677}{134750364710796692975428197678665525523666976000} a^{14} + \frac{78048145586085562451513747780951401460799}{269500729421593385950856395357331051047333952000} a^{13} - \frac{116457600312351599823367277546640233200971}{134750364710796692975428197678665525523666976000} a^{12} - \frac{1384500677670989319652249934234001340375911}{269500729421593385950856395357331051047333952000} a^{11} - \frac{11438323790580723359485899085594889136706443}{134750364710796692975428197678665525523666976000} a^{10} + \frac{79889018189682359720307341312273602157345851}{134750364710796692975428197678665525523666976000} a^{9} + \frac{9976803187671984955178795467275505884379167}{1418424891692596768162402080828058163407020800} a^{8} + \frac{429400148746066297917207731210735373202711353}{14184248916925967681624020808280581634070208000} a^{7} - \frac{15065000844082298724276654880770314369794833}{886515557307872980101501300517536352129388000} a^{6} - \frac{3470404240933262858740695553589010835656501}{11081444466348412251268766256469204401617350} a^{5} - \frac{187890797034956283000573455214965550155380959}{221628889326968245025375325129384088032347000} a^{4} + \frac{107029183858432002085888796207972309521108731}{443257778653936490050750650258768176064694000} a^{3} + \frac{514386005353268047709860543257932090909659299}{110814444663484122512687662564692044016173500} a^{2} + \frac{2555138160742143258316827183140780664894515307}{221628889326968245025375325129384088032347000} a + \frac{462578246764665406301147407130536173882675411}{55407222331742061256343831282346022008086750} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[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)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 6840674555320000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times D_9$ (as 18T50):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 18 conjugacy class representatives for $S_3\times D_9$
Character table for $S_3\times D_9$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.3.146205.1, 6.0.64127706075.3

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{9}$ R R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ $18$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
3Data not computed
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$19$19.9.8.9$x^{9} - 4864$$9$$1$$8$$C_9$$[\ ]_{9}$
19.9.8.1$x^{9} + 76$$9$$1$$8$$C_9$$[\ ]_{9}$