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Properties

Label 18.0.19084030575...6496.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{33}\cdot 3^{27}\cdot 79^{15}$
Root discriminant $706.20$
Ramified primes $2, 3, 79$
Class number $2616929280$ (GRH)
Class group $[2, 2, 2, 2, 4, 12, 3407460]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4363300165325176, 3660812605939536, 1395378492366564, 273825317586768, 22616881415904, -1796953824576, -580604160344, -33229098816, 7033272066, 1152845420, 26855961, -9046764, 1078822, 355500, 28422, -1580, -276, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 276*x^16 - 1580*x^15 + 28422*x^14 + 355500*x^13 + 1078822*x^12 - 9046764*x^11 + 26855961*x^10 + 1152845420*x^9 + 7033272066*x^8 - 33229098816*x^7 - 580604160344*x^6 - 1796953824576*x^5 + 22616881415904*x^4 + 273825317586768*x^3 + 1395378492366564*x^2 + 3660812605939536*x + 4363300165325176)
 
gp: K = bnfinit(x^18 - 276*x^16 - 1580*x^15 + 28422*x^14 + 355500*x^13 + 1078822*x^12 - 9046764*x^11 + 26855961*x^10 + 1152845420*x^9 + 7033272066*x^8 - 33229098816*x^7 - 580604160344*x^6 - 1796953824576*x^5 + 22616881415904*x^4 + 273825317586768*x^3 + 1395378492366564*x^2 + 3660812605939536*x + 4363300165325176, 1)
 

Normalized defining polynomial

\( x^{18} - 276 x^{16} - 1580 x^{15} + 28422 x^{14} + 355500 x^{13} + 1078822 x^{12} - 9046764 x^{11} + 26855961 x^{10} + 1152845420 x^{9} + 7033272066 x^{8} - 33229098816 x^{7} - 580604160344 x^{6} - 1796953824576 x^{5} + 22616881415904 x^{4} + 273825317586768 x^{3} + 1395378492366564 x^{2} + 3660812605939536 x + 4363300165325176 \)

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:  \(-1908403057576682162285171000138631615439273507946496=-\,2^{33}\cdot 3^{27}\cdot 79^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $706.20$
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:  $2, 3, 79$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{6} a^{9} + \frac{1}{3} a^{6} - \frac{1}{2} a^{5} - \frac{1}{3}$, $\frac{1}{6} a^{10} + \frac{1}{3} a^{7} - \frac{1}{2} a^{6} - \frac{1}{3} a$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{24} a^{12} - \frac{1}{12} a^{11} - \frac{1}{24} a^{10} - \frac{1}{24} a^{8} - \frac{1}{12} a^{7} + \frac{5}{24} a^{6} - \frac{1}{4} a^{4} - \frac{1}{3} a^{3} - \frac{1}{12} a^{2} + \frac{1}{3} a + \frac{1}{6}$, $\frac{1}{48} a^{13} + \frac{1}{16} a^{11} + \frac{1}{24} a^{10} + \frac{1}{16} a^{9} - \frac{1}{4} a^{8} - \frac{5}{16} a^{7} + \frac{1}{8} a^{6} - \frac{3}{8} a^{5} - \frac{5}{12} a^{4} + \frac{1}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{48} a^{14} - \frac{1}{48} a^{12} + \frac{1}{24} a^{11} - \frac{1}{48} a^{10} - \frac{1}{12} a^{9} - \frac{1}{16} a^{8} + \frac{11}{24} a^{7} + \frac{1}{24} a^{6} + \frac{1}{12} a^{5} + \frac{1}{8} a^{4} + \frac{5}{12} a^{3} + \frac{1}{4} a^{2} + \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{1763280} a^{15} - \frac{1}{2790} a^{14} - \frac{4483}{881640} a^{13} + \frac{107}{11160} a^{12} + \frac{5233}{73470} a^{11} - \frac{353}{11160} a^{10} - \frac{1631}{28440} a^{9} - \frac{2369}{11160} a^{8} + \frac{94823}{587760} a^{7} - \frac{1813}{3720} a^{6} + \frac{29992}{110205} a^{5} - \frac{617}{1395} a^{4} - \frac{195809}{881640} a^{3} - \frac{761}{5580} a^{2} + \frac{67}{372} a - \frac{13}{90}$, $\frac{1}{1763280} a^{16} - \frac{287}{117552} a^{14} + \frac{2}{465} a^{13} - \frac{26641}{1763280} a^{12} + \frac{149}{2232} a^{11} + \frac{6599}{117552} a^{10} - \frac{38}{465} a^{9} + \frac{11533}{176328} a^{8} - \frac{1031}{3720} a^{7} - \frac{250891}{881640} a^{6} - \frac{259}{930} a^{5} - \frac{19181}{440820} a^{4} - \frac{1867}{5580} a^{3} - \frac{731}{2790} a^{2} - \frac{883}{2790} a + \frac{17}{45}$, $\frac{1}{911268622585164750000518064708733732418048326108130033749309502921976360069775284747820737719810821044960} a^{17} - \frac{23904742670502917517730174513627686295252106251731799905871421442656638340610540224771749064154061}{101252069176129416666724229412081525824227591789792225972145500324664040007752809416424526413312313449440} a^{16} + \frac{25839523270689973469327467336126845609653855883033336076427126674474194844844430357363012850110481}{303756207528388250000172688236244577472682775369376677916436500973992120023258428249273579239936940348320} a^{15} + \frac{490434041062300784373795900537003401944874808266757167771593303318212872274876331308618141127825276157}{101252069176129416666724229412081525824227591789792225972145500324664040007752809416424526413312313449440} a^{14} + \frac{1482959386066278176516050013755008853286804937479733317287061667188806704370937734884743060398235003017}{182253724517032950000103612941746746483609665221626006749861900584395272013955056949564147543962164208992} a^{13} - \frac{15340751884327817090720652076904784398456068543153878190145533488860852390822770446482639590520511524083}{911268622585164750000518064708733732418048326108130033749309502921976360069775284747820737719810821044960} a^{12} + \frac{6313852445962004595500859894156550420775918685106729649103830991988490956163285721009913018085036189649}{101252069176129416666724229412081525824227591789792225972145500324664040007752809416424526413312313449440} a^{11} - \frac{8907163421215936488593144480326885497135107713380910035488739796593553187639756311607524786044559955201}{303756207528388250000172688236244577472682775369376677916436500973992120023258428249273579239936940348320} a^{10} + \frac{11847908978730763303514233081744471321082428655577796378100974971243492527588512163566779212022904759087}{227817155646291187500129516177183433104512081527032508437327375730494090017443821186955184429952705261240} a^{9} + \frac{1305646333576819574386880872739550386612200000220752453335728438399943787580997147106041004333422171257}{30375620752838825000017268823624457747268277536937667791643650097399212002325842824927357923993694034832} a^{8} + \frac{13240149556693534884348901363082761747899720724552045856113928954294198269506446955785141060328773856443}{455634311292582375000259032354366866209024163054065016874654751460988180034887642373910368859905410522480} a^{7} - \frac{3419059501293037093837906948403996429577984468417568628887937617445090089964706179267464553915199682061}{75939051882097062500043172059061144368170693842344169479109125243498030005814607062318394809984235087080} a^{6} + \frac{15541365146409909526348041500754520723847503217704815201149046475458670174096532470982259821764893139481}{113908577823145593750064758088591716552256040763516254218663687865247045008721910593477592214976352630620} a^{5} - \frac{268019113524133102346881271731812674127738588219550677024682808128058511929370184570847673444709032479}{1469788100943814112904061394691506020029110203400209731853725004712865096886734330238420544709372292008} a^{4} + \frac{56670423719212896769952821403013647508712543205522605688221197632601711878343209121718477137044758995499}{113908577823145593750064758088591716552256040763516254218663687865247045008721910593477592214976352630620} a^{3} - \frac{1328406828413776414750009700329426673480792195968394585706739404457193615882982479103404250180040686641}{2883761463877103639242145774394727001322937740848512765029460452284735316676504065657660562404464623560} a^{2} + \frac{292657769272880816334613237733388890052044353992582398688659791501655069306504771778400568665326869573}{720940365969275909810536443598681750330734435212128191257365113071183829169126016414415140601116155890} a - \frac{225985798610884376622453928313484418894536257164660157695663155124719462508029394718749834726968229}{3100818778362477031443167499349168818626814775105927704332753174499715394275810823287807056348886692}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{12}\times C_{3407460}$, which has order $2616929280$ (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:  \( -1 \) (order $2$)
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:  \( 32038743174.66701 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-474}) \), 3.3.505521.1, 3.3.316.1, 6.0.31009638963976704.2, 6.0.27263084544.4, 9.9.653167654112852807616.1

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

Sibling fields

Galois closure: data not computed
Degree 12 sibling: data not computed
Degree 18 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 R R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$2$2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.4.8.1$x^{4} + 2 x^{2} + 4 x + 10$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.1$x^{4} + 2 x^{2} + 4 x + 10$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.1$x^{4} + 2 x^{2} + 4 x + 10$$4$$1$$8$$C_2^2$$[2, 3]$
3Data not computed
$79$79.6.5.2$x^{6} - 316$$6$$1$$5$$C_6$$[\ ]_{6}$
79.6.5.2$x^{6} - 316$$6$$1$$5$$C_6$$[\ ]_{6}$
79.6.5.2$x^{6} - 316$$6$$1$$5$$C_6$$[\ ]_{6}$