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Properties

Label 18.0.31718053554...7888.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 7^{15}\cdot 19^{6}\cdot 211^{14}$
Root discriminant $1377.09$
Ramified primes $2, 7, 19, 211$
Class number $15104931030$ (GRH)
Class group $[3, 3, 3, 9, 9, 6906690]$ (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![783773403510784, 674017524612704, 719202938309504, -46074585055336, -74670398987968, -27873163776444, 9217764115764, 1634195356934, -388634200160, -37726851253, 8041909773, 365710044, -79528672, -1518902, 399126, 2510, -984, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 984*x^16 + 2510*x^15 + 399126*x^14 - 1518902*x^13 - 79528672*x^12 + 365710044*x^11 + 8041909773*x^10 - 37726851253*x^9 - 388634200160*x^8 + 1634195356934*x^7 + 9217764115764*x^6 - 27873163776444*x^5 - 74670398987968*x^4 - 46074585055336*x^3 + 719202938309504*x^2 + 674017524612704*x + 783773403510784)
 
gp: K = bnfinit(x^18 - x^17 - 984*x^16 + 2510*x^15 + 399126*x^14 - 1518902*x^13 - 79528672*x^12 + 365710044*x^11 + 8041909773*x^10 - 37726851253*x^9 - 388634200160*x^8 + 1634195356934*x^7 + 9217764115764*x^6 - 27873163776444*x^5 - 74670398987968*x^4 - 46074585055336*x^3 + 719202938309504*x^2 + 674017524612704*x + 783773403510784, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} - 984 x^{16} + 2510 x^{15} + 399126 x^{14} - 1518902 x^{13} - 79528672 x^{12} + 365710044 x^{11} + 8041909773 x^{10} - 37726851253 x^{9} - 388634200160 x^{8} + 1634195356934 x^{7} + 9217764115764 x^{6} - 27873163776444 x^{5} - 74670398987968 x^{4} - 46074585055336 x^{3} + 719202938309504 x^{2} + 674017524612704 x + 783773403510784 \)

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:  \(-317180535547366983497681454181403157145875143430046117888=-\,2^{12}\cdot 7^{15}\cdot 19^{6}\cdot 211^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1377.09$
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:  $2, 7, 19, 211$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4}$, $\frac{1}{16} a^{7} + \frac{1}{16} a^{6} + \frac{1}{16} a^{5} + \frac{1}{16} a^{4} + \frac{1}{8} a^{3} + \frac{3}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{16} a^{8} + \frac{1}{16} a^{4} - \frac{1}{4} a^{3} + \frac{3}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{16} a^{9} + \frac{1}{16} a^{5} - \frac{1}{4} a^{4} - \frac{1}{8} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{32} a^{10} + \frac{1}{32} a^{6} - \frac{1}{8} a^{5} - \frac{1}{16} a^{4} + \frac{1}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{64} a^{11} - \frac{1}{32} a^{8} - \frac{1}{64} a^{7} + \frac{1}{32} a^{6} - \frac{1}{16} a^{5} + \frac{1}{8} a^{4} + \frac{1}{16} a^{3} + \frac{3}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{256} a^{12} - \frac{1}{256} a^{11} + \frac{1}{128} a^{10} - \frac{3}{128} a^{9} + \frac{1}{256} a^{8} + \frac{7}{256} a^{7} - \frac{1}{16} a^{6} - \frac{7}{64} a^{5} + \frac{15}{64} a^{4} - \frac{13}{64} a^{3} - \frac{3}{16} a^{2} + \frac{5}{16} a$, $\frac{1}{256} a^{13} + \frac{1}{256} a^{11} - \frac{1}{64} a^{10} - \frac{5}{256} a^{9} - \frac{1}{32} a^{8} + \frac{7}{256} a^{7} - \frac{7}{64} a^{6} - \frac{1}{16} a^{5} + \frac{1}{32} a^{4} + \frac{15}{64} a^{3} - \frac{3}{8} a^{2} + \frac{5}{16} a$, $\frac{1}{256} a^{14} + \frac{1}{256} a^{11} + \frac{1}{256} a^{10} - \frac{1}{128} a^{9} - \frac{1}{128} a^{8} - \frac{7}{256} a^{7} - \frac{1}{16} a^{6} + \frac{5}{64} a^{5} - \frac{1}{16} a^{4} - \frac{15}{64} a^{3} + \frac{1}{8} a^{2} + \frac{3}{16} a$, $\frac{1}{82432} a^{15} + \frac{33}{41216} a^{14} - \frac{121}{82432} a^{13} - \frac{5}{5888} a^{12} + \frac{381}{82432} a^{11} - \frac{15}{5888} a^{10} + \frac{97}{82432} a^{9} + \frac{1}{5888} a^{8} + \frac{251}{41216} a^{7} + \frac{47}{736} a^{6} - \frac{1121}{20608} a^{5} - \frac{259}{1472} a^{4} - \frac{1919}{10304} a^{3} + \frac{65}{368} a^{2} + \frac{433}{2576} a + \frac{29}{161}$, $\frac{1}{866195456} a^{16} - \frac{2283}{866195456} a^{15} + \frac{236907}{123742208} a^{14} - \frac{245209}{866195456} a^{13} - \frac{618293}{866195456} a^{12} - \frac{2722851}{866195456} a^{11} - \frac{6461813}{866195456} a^{10} - \frac{8276551}{866195456} a^{9} + \frac{615}{588448} a^{8} + \frac{12871989}{433097728} a^{7} - \frac{20676181}{216548864} a^{6} - \frac{13202493}{216548864} a^{5} + \frac{13430609}{54137216} a^{4} - \frac{17137221}{108274432} a^{3} + \frac{5395771}{13534304} a^{2} + \frac{9224879}{27068608} a - \frac{34700}{422947}$, $\frac{1}{4404494955201184495356647409304090297573990916458197262310478424733455578966339563103298523565221306368} a^{17} + \frac{1033286861991797275049172680167967534703324557432796374935143264961622668432759866315543367501}{2202247477600592247678323704652045148786995458229098631155239212366727789483169781551649261782610653184} a^{16} - \frac{3454218290300073449878529902304709901123111721958690579176546248002613770177104285738611814325653}{2202247477600592247678323704652045148786995458229098631155239212366727789483169781551649261782610653184} a^{15} - \frac{325376323530162042057974400692224887188172091383341196879858435712505586525665128593544527105669155}{275280934700074030959790463081505643598374432278637328894404901545840973685396222693956157722826331648} a^{14} - \frac{1997914725545403118171600964274371358288304340507698289126693822613511095520302856952032195419799}{8502886013901900570186578010239556559023148487371037185927564526512462507657026183597101396844056576} a^{13} - \frac{1204466795726978863715472999975617710888947779366682837152503363376106965224714431959067410903168821}{1101123738800296123839161852326022574393497729114549315577619606183363894741584890775824630891305326592} a^{12} + \frac{134421516237213656157254638038372391041648066132522423650474207361406495727661472067892607252071203}{157303391257185160548451693189431796341928247016364187939659943740480556391654984396546375841615046656} a^{11} + \frac{105660550478138786881710796964418164713034913087124605502598158070817724865986151122084664503523621}{14880050524328325997826511517919223978290509852899315075373237921396809388399795821294927444477099008} a^{10} + \frac{8822066586828585802841667819622700416257284945474085301445929658061265315550114317950829608993048419}{629213565028740642193806772757727185367712988065456751758639774961922225566619937586185503366460186624} a^{9} - \frac{20495627383764008688329581039640634792865018661438901990730559903419730185358243614146199218133352007}{2202247477600592247678323704652045148786995458229098631155239212366727789483169781551649261782610653184} a^{8} - \frac{6560259570899421557750078595947653522021110426759758015324679311561815226306621199305998285243879051}{314606782514370321096903386378863592683856494032728375879319887480961112783309968793092751683230093312} a^{7} + \frac{1800878071185946729314416028994978086810664404716415528501861106666510393995999042583291218213248673}{550561869400148061919580926163011287196748864557274657788809803091681947370792445387912315445652663296} a^{6} + \frac{8071035822908790386502347713287258724443585135904052282266042062821219654706006316757968959483147573}{157303391257185160548451693189431796341928247016364187939659943740480556391654984396546375841615046656} a^{5} + \frac{216137384426656168812080709651338970099580434552718069161280278100067492706234004545828418605312433}{4872228932744673114332574567814259178732290836790041219369998257448512808591083587503648809253563392} a^{4} - \frac{1398714919938789073675810866902980971557695544346642527780439227650429823080142953337115695427705465}{14880050524328325997826511517919223978290509852899315075373237921396809388399795821294927444477099008} a^{3} - \frac{48864683459841225716086227447287260818347259040348056044320002104773094483210278401963650180330577631}{137640467350037015479895231540752821799187216139318664447202450772920486842698111346978078861413165824} a^{2} + \frac{37696700183444854435756632299260279975897594746430175190825013754713271956110986316448354298512857175}{137640467350037015479895231540752821799187216139318664447202450772920486842698111346978078861413165824} a - \frac{491635555056436475161790629747211147109258183915771130928381811727392283974262077627042058146884841}{2150632302344328366873362992824262840612300252176854131987538293326882606917157989796532482209580716}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{9}\times C_{9}\times C_{6906690}$, which has order $15104931030$ (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:  \( 182929425339792.53 \) (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

\(\Q(\sqrt{-7}) \), 3.3.2181529.1, 3.3.112252.1, 6.0.88203580528.1, 6.0.33313481444887.3, 9.9.6731382107576072178358049728.1

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
$7$7.6.5.6$x^{6} + 224$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.6$x^{6} + 224$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.6$x^{6} + 224$$6$$1$$5$$C_6$$[\ ]_{6}$
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
211Data not computed