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Properties

Label 18.0.19579865218...9787.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{31}\cdot 7^{12}\cdot 73^{12}$
Root discriminant $423.96$
Ramified primes $3, 7, 73$
Class number $2387469168$ (GRH)
Class group $[3, 3, 3, 3, 9, 684, 4788]$ (GRH)
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![17746624531, -114085331286, 319438195035, -479154240201, 410694739059, -205328532297, 68414239629, 41336523, -35056719, 22236314, -10201200, 3137505, -521793, -861, 411, -156, 45, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 45*x^16 - 156*x^15 + 411*x^14 - 861*x^13 - 521793*x^12 + 3137505*x^11 - 10201200*x^10 + 22236314*x^9 - 35056719*x^8 + 41336523*x^7 + 68414239629*x^6 - 205328532297*x^5 + 410694739059*x^4 - 479154240201*x^3 + 319438195035*x^2 - 114085331286*x + 17746624531)
 
gp: K = bnfinit(x^18 - 9*x^17 + 45*x^16 - 156*x^15 + 411*x^14 - 861*x^13 - 521793*x^12 + 3137505*x^11 - 10201200*x^10 + 22236314*x^9 - 35056719*x^8 + 41336523*x^7 + 68414239629*x^6 - 205328532297*x^5 + 410694739059*x^4 - 479154240201*x^3 + 319438195035*x^2 - 114085331286*x + 17746624531, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 45 x^{16} - 156 x^{15} + 411 x^{14} - 861 x^{13} - 521793 x^{12} + 3137505 x^{11} - 10201200 x^{10} + 22236314 x^{9} - 35056719 x^{8} + 41336523 x^{7} + 68414239629 x^{6} - 205328532297 x^{5} + 410694739059 x^{4} - 479154240201 x^{3} + 319438195035 x^{2} - 114085331286 x + 17746624531 \)

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:  \(-195798652181541517842155358339748719235115999787=-\,3^{31}\cdot 7^{12}\cdot 73^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $423.96$
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:  $3, 7, 73$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{8} a^{6} - \frac{3}{8} a^{5} - \frac{1}{4} a^{4} + \frac{1}{8} a^{3} - \frac{3}{8} a^{2} - \frac{1}{4} a + \frac{3}{8}$, $\frac{1}{8} a^{7} - \frac{3}{8} a^{5} + \frac{3}{8} a^{4} - \frac{3}{8} a^{2} - \frac{3}{8} a + \frac{1}{8}$, $\frac{1}{2728} a^{8} - \frac{1}{682} a^{7} + \frac{5}{1364} a^{6} - \frac{2}{341} a^{5} - \frac{247}{1364} a^{4} + \frac{505}{1364} a^{3} - \frac{551}{1364} a^{2} + \frac{595}{2728} a - \frac{85}{2728}$, $\frac{1}{2728} a^{9} - \frac{3}{1364} a^{7} + \frac{3}{341} a^{6} - \frac{9}{44} a^{5} - \frac{483}{1364} a^{4} + \frac{105}{1364} a^{3} - \frac{35}{88} a^{2} - \frac{433}{2728} a - \frac{85}{682}$, $\frac{1}{2728} a^{10} - \frac{157}{2728} a^{6} + \frac{643}{2728} a^{5} - \frac{177}{682} a^{4} - \frac{35}{682} a^{3} + \frac{29}{682} a^{2} - \frac{45}{682} a + \frac{513}{2728}$, $\frac{1}{2728} a^{11} - \frac{157}{2728} a^{7} - \frac{39}{2728} a^{6} + \frac{669}{1364} a^{5} + \frac{153}{341} a^{4} - \frac{283}{1364} a^{3} - \frac{431}{1364} a^{2} - \frac{851}{2728} a + \frac{1}{4}$, $\frac{1}{21824} a^{12} + \frac{1}{10912} a^{11} - \frac{3}{21824} a^{10} - \frac{1}{10912} a^{9} - \frac{655}{10912} a^{7} + \frac{1207}{21824} a^{6} - \frac{7}{341} a^{5} + \frac{7833}{21824} a^{4} + \frac{2245}{10912} a^{3} - \frac{9}{992} a^{2} - \frac{1187}{5456} a + \frac{5233}{21824}$, $\frac{1}{21824} a^{13} + \frac{1}{21824} a^{11} - \frac{1}{5456} a^{10} - \frac{1}{5456} a^{9} + \frac{1}{10912} a^{8} + \frac{9}{1984} a^{7} + \frac{49}{10912} a^{6} + \frac{489}{21824} a^{5} + \frac{5}{22} a^{4} - \frac{4749}{10912} a^{3} - \frac{97}{341} a^{2} - \frac{383}{21824} a + \frac{2423}{10912}$, $\frac{1}{21824} a^{14} + \frac{1}{10912} a^{11} - \frac{1}{21824} a^{10} - \frac{1}{5456} a^{9} + \frac{3}{21824} a^{8} + \frac{73}{2728} a^{7} + \frac{273}{10912} a^{6} - \frac{987}{2728} a^{5} - \frac{1491}{21824} a^{4} - \frac{1009}{10912} a^{3} - \frac{9913}{21824} a^{2} + \frac{4573}{10912} a - \frac{2537}{21824}$, $\frac{1}{21824} a^{15} + \frac{3}{21824} a^{11} + \frac{1}{10912} a^{10} - \frac{1}{21824} a^{9} - \frac{581}{10912} a^{7} - \frac{13}{992} a^{6} - \frac{8819}{21824} a^{5} - \frac{19}{496} a^{4} - \frac{9325}{21824} a^{3} - \frac{5369}{10912} a^{2} + \frac{10775}{21824} a + \frac{1851}{10912}$, $\frac{1}{34871733099076199569118337088} a^{16} - \frac{1}{4358966637384524946139792136} a^{15} + \frac{86353541753520960887455}{34871733099076199569118337088} a^{14} - \frac{604474792274646726212045}{34871733099076199569118337088} a^{13} + \frac{31436667449989030956765}{17435866549538099784559168544} a^{12} - \frac{5301961074593463705328455}{34871733099076199569118337088} a^{11} + \frac{5010890579676544518097551}{34871733099076199569118337088} a^{10} - \frac{2886583928023194418366891}{17435866549538099784559168544} a^{9} - \frac{3825403516658488149723095}{34871733099076199569118337088} a^{8} + \frac{2163842247604243645807874089}{34871733099076199569118337088} a^{7} + \frac{24417949192868223685558649}{17435866549538099784559168544} a^{6} + \frac{1030407508626590318200255919}{34871733099076199569118337088} a^{5} - \frac{1181434010264751871975422179}{34871733099076199569118337088} a^{4} + \frac{109973354638012113026637149}{1089741659346131236534948034} a^{3} + \frac{731908960267480045380598033}{1585078777230736344050833504} a^{2} + \frac{3382019766067921227470580387}{34871733099076199569118337088} a + \frac{67678307211312189557490907}{260236814172210444545659232}$, $\frac{1}{324393285873896473281935826329344448} a^{17} + \frac{4651227}{324393285873896473281935826329344448} a^{16} - \frac{2904005687467785807321359191715}{162196642936948236640967913164672224} a^{15} + \frac{3400790669578004643005013307675}{162196642936948236640967913164672224} a^{14} + \frac{5849968118778114782420439832107}{324393285873896473281935826329344448} a^{13} + \frac{522231427910787345219719428687}{324393285873896473281935826329344448} a^{12} + \frac{25724532788346326854999129003119}{324393285873896473281935826329344448} a^{11} - \frac{40317803627473378378861494439993}{324393285873896473281935826329344448} a^{10} - \frac{4773184323865087327935036508639}{81098321468474118320483956582336112} a^{9} + \frac{18779206846762662329710995102699}{162196642936948236640967913164672224} a^{8} + \frac{19660507606083875724943075156088303}{324393285873896473281935826329344448} a^{7} - \frac{14717941336354080168311611624679617}{324393285873896473281935826329344448} a^{6} - \frac{46682557333804820348168512591394511}{324393285873896473281935826329344448} a^{5} + \frac{54746255135200910819274766771397665}{324393285873896473281935826329344448} a^{4} - \frac{153302634023410285570829745226871373}{324393285873896473281935826329344448} a^{3} - \frac{58255659377778650047633119509175759}{324393285873896473281935826329344448} a^{2} - \frac{27965836113307764438072824895543295}{81098321468474118320483956582336112} a + \frac{446948160502018190499443606489595}{2420845416969376666283103181562272}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{9}\times C_{684}\times C_{4788}$, which has order $2387469168$ (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:  \( \frac{599785579809883182727302}{5068645091779632395030247286396007} a^{17} - \frac{5098177428384007053182067}{5068645091779632395030247286396007} a^{16} + \frac{24524563487554975333099844}{5068645091779632395030247286396007} a^{15} - \frac{1223442949089286178128470}{75651419280293020821346974423821} a^{14} + \frac{18953486704401626649852318}{460785917434512035911840662399637} a^{13} - \frac{421241786587116428682605870}{5068645091779632395030247286396007} a^{12} - \frac{313153575829049141579078830998}{5068645091779632395030247286396007} a^{11} + \frac{156837753721183764079542201144}{460785917434512035911840662399637} a^{10} - \frac{5299457399030003949230267105024}{5068645091779632395030247286396007} a^{9} + \frac{10905166909489505300311216710165}{5068645091779632395030247286396007} a^{8} - \frac{1468420402425809849293162535418}{460785917434512035911840662399637} a^{7} + \frac{17724068567758616707224922694282}{5068645091779632395030247286396007} a^{6} + \frac{41041465648513882234748389871902182}{5068645091779632395030247286396007} a^{5} - \frac{102631158062643455255134818023031423}{5068645091779632395030247286396007} a^{4} + \frac{200714407987018019588187473094233002}{5068645091779632395030247286396007} a^{3} - \frac{198436273240040455847098616290399296}{5068645091779632395030247286396007} a^{2} + \frac{109482525260625542239538693815543116}{5068645091779632395030247286396007} a - \frac{336648592853452238496196971234703}{75651419280293020821346974423821} \) (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:  \( 49147979.00484934 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

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

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
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

\(\Q(\sqrt{-3}) \), 3.1.7050267.2 x3, 3.3.431649.2, 6.0.149118794313867.1, 6.0.251842587507.2 x2, 6.0.558962577603.2, 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$73$73.3.2.1$x^{3} - 73$$3$$1$$2$$C_3$$[\ ]_{3}$
73.3.2.1$x^{3} - 73$$3$$1$$2$$C_3$$[\ ]_{3}$
73.3.2.1$x^{3} - 73$$3$$1$$2$$C_3$$[\ ]_{3}$
73.3.2.1$x^{3} - 73$$3$$1$$2$$C_3$$[\ ]_{3}$
73.3.2.1$x^{3} - 73$$3$$1$$2$$C_3$$[\ ]_{3}$
73.3.2.1$x^{3} - 73$$3$$1$$2$$C_3$$[\ ]_{3}$