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Properties

Label 18.0.85595773557...7712.4
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{39}\cdot 13^{12}\cdot 19^{12}$
Root discriminant $675.43$
Ramified primes $2, 3, 13, 19$
Class number $1191969675$ (GRH)
Class group $[3, 3, 3, 3, 585, 25155]$ (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![65006631218368, 51445398142656, 16521840365184, 3330748799808, 780864960960, 193086664128, -18882531552, -19766135520, -423343152, 949735792, 86941944, -20110392, -2674572, 202284, 40104, -708, -306, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 306*x^16 - 708*x^15 + 40104*x^14 + 202284*x^13 - 2674572*x^12 - 20110392*x^11 + 86941944*x^10 + 949735792*x^9 - 423343152*x^8 - 19766135520*x^7 - 18882531552*x^6 + 193086664128*x^5 + 780864960960*x^4 + 3330748799808*x^3 + 16521840365184*x^2 + 51445398142656*x + 65006631218368)
 
gp: K = bnfinit(x^18 - 306*x^16 - 708*x^15 + 40104*x^14 + 202284*x^13 - 2674572*x^12 - 20110392*x^11 + 86941944*x^10 + 949735792*x^9 - 423343152*x^8 - 19766135520*x^7 - 18882531552*x^6 + 193086664128*x^5 + 780864960960*x^4 + 3330748799808*x^3 + 16521840365184*x^2 + 51445398142656*x + 65006631218368, 1)
 

Normalized defining polynomial

\( x^{18} - 306 x^{16} - 708 x^{15} + 40104 x^{14} + 202284 x^{13} - 2674572 x^{12} - 20110392 x^{11} + 86941944 x^{10} + 949735792 x^{9} - 423343152 x^{8} - 19766135520 x^{7} - 18882531552 x^{6} + 193086664128 x^{5} + 780864960960 x^{4} + 3330748799808 x^{3} + 16521840365184 x^{2} + 51445398142656 x + 65006631218368 \)

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:  \(-855957735577256641291889992748058377744964357107712=-\,2^{12}\cdot 3^{39}\cdot 13^{12}\cdot 19^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $675.43$
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:  $2, 3, 13, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{4} a^{6}$, $\frac{1}{28} a^{7} - \frac{2}{7} a$, $\frac{1}{28} a^{8} - \frac{2}{7} a^{2}$, $\frac{1}{56} a^{9} - \frac{1}{7} a^{3}$, $\frac{1}{56} a^{10} - \frac{1}{7} a^{4}$, $\frac{1}{112} a^{11} - \frac{1}{56} a^{8} - \frac{1}{14} a^{5} - \frac{5}{14} a^{2}$, $\frac{1}{112} a^{12} - \frac{1}{14} a^{6}$, $\frac{1}{784} a^{13} - \frac{1}{392} a^{12} - \frac{3}{784} a^{11} + \frac{3}{392} a^{10} + \frac{1}{392} a^{9} + \frac{5}{392} a^{8} - \frac{1}{98} a^{7} + \frac{9}{98} a^{6} - \frac{11}{98} a^{5} + \frac{11}{49} a^{4} - \frac{9}{98} a^{3} - \frac{45}{98} a^{2} - \frac{2}{7} a$, $\frac{1}{1568} a^{14} + \frac{3}{392} a^{8} - \frac{1}{4} a^{5} + \frac{39}{98} a^{2}$, $\frac{1}{923552} a^{15} - \frac{3}{24304} a^{14} - \frac{127}{230888} a^{13} + \frac{179}{57722} a^{12} - \frac{99}{461776} a^{11} + \frac{134}{28861} a^{10} - \frac{1567}{230888} a^{9} + \frac{1055}{230888} a^{8} + \frac{557}{115444} a^{7} - \frac{3075}{28861} a^{6} + \frac{10221}{57722} a^{5} - \frac{5951}{28861} a^{4} + \frac{173}{1519} a^{3} - \frac{28681}{57722} a^{2} + \frac{89}{217} a + \frac{11}{31}$, $\frac{1}{2004348887072} a^{16} + \frac{985783}{2004348887072} a^{15} - \frac{530993299}{2004348887072} a^{14} + \frac{53103269}{125271805442} a^{13} + \frac{13958823}{13186505836} a^{12} - \frac{222783419}{250543610884} a^{11} - \frac{267994653}{125271805442} a^{10} - \frac{1070253305}{501087221768} a^{9} + \frac{4591705277}{501087221768} a^{8} - \frac{2079531071}{125271805442} a^{7} + \frac{6696607122}{62635902721} a^{6} + \frac{4109582419}{250543610884} a^{5} - \frac{25891203927}{125271805442} a^{4} - \frac{565912619}{2556567458} a^{3} - \frac{61344994369}{125271805442} a^{2} + \frac{8345253}{67278091} a + \frac{30848950}{67278091}$, $\frac{1}{7723609694707715702313646689516396153734318209148862804632984591122434202816} a^{17} - \frac{236234982430739575149504110944437760704726325804168351378064845}{965451211838464462789205836189549519216789776143607850579123073890304275352} a^{16} - \frac{16943648443109075498290181872363980405717104560383508191963911584709}{482725605919232231394602918094774759608394888071803925289561536945152137676} a^{15} + \frac{524956225056400720274589721946579756660940311398260304113349489564208141}{1930902423676928925578411672379099038433579552287215701158246147780608550704} a^{14} + \frac{508793400072521347160176707276520545537516222358135070512594219621969907}{965451211838464462789205836189549519216789776143607850579123073890304275352} a^{13} + \frac{190940157095928282048630129791888501101467890852854163136878118566201699}{50813221675708655936273991378397343116673146112821465819953845994226540808} a^{12} + \frac{8561776849638847756828762620862632772239077649288635682168063760725881295}{1930902423676928925578411672379099038433579552287215701158246147780608550704} a^{11} - \frac{8130871881170049340527023283639601466668674536488004907437941883974972337}{965451211838464462789205836189549519216789776143607850579123073890304275352} a^{10} - \frac{2121365068614361255559669666421637181089458652513339208634911395345135125}{965451211838464462789205836189549519216789776143607850579123073890304275352} a^{9} + \frac{5475723094415411841835666816896388053240171476717874146877843798072372855}{965451211838464462789205836189549519216789776143607850579123073890304275352} a^{8} - \frac{94352612321258261504878312314450276708931969513759716971043774816559885}{7785896869665035990235530937012496122716046581803289117573573176534711898} a^{7} + \frac{11897722444893869925677809900327397386791483194122329728939117697525271996}{120681401479808057848650729523693689902098722017950981322390384236288034419} a^{6} - \frac{265122103808977952424188549230536229187582733708098664881419657110111735}{15571793739330071980471061874024992245432093163606578235147146353069423796} a^{5} + \frac{336045919740242855044352391508293817296863871565452999831438288921124913}{3892948434832517995117765468506248061358023290901644558786786588267355949} a^{4} - \frac{9711717742432094631156177837065987780191263608577907777079877174405375759}{120681401479808057848650729523693689902098722017950981322390384236288034419} a^{3} - \frac{147840542346894241923286109984973847576746886313624462675841081470372584}{351840820640839818800730989864996180472591026291402277907843685820081733} a^{2} - \frac{5579307168221085053372802475712005490666799303438305955019882534971707}{129625565499256775347637733108156487542533536002095576071310831617924849} a - \frac{832733865499150123133717934332299851742636538913721864579382954332348}{2645419704066464803013014961390948725357827265348889307577772073835201}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{585}\times C_{25155}$, which has order $1191969675$ (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:  \( \frac{10203222017368779600350627678945627847}{29639200408387971954859467884478072427342791840064} a^{17} - \frac{15691197148483599459506455814132737419}{14819600204193985977429733942239036213671395920032} a^{16} - \frac{1511341573833577333893339022275089531823}{14819600204193985977429733942239036213671395920032} a^{15} + \frac{1014731737886886148358534725355628720135}{14819600204193985977429733942239036213671395920032} a^{14} + \frac{50262791295472163718855725868250709573919}{3704900051048496494357433485559759053417848980008} a^{13} + \frac{209643575320323227917038124223810075894169}{7409800102096992988714866971119518106835697960016} a^{12} - \frac{7434895914242145567529465221850125016882503}{7409800102096992988714866971119518106835697960016} a^{11} - \frac{14368832151703194141700566312672359828602863}{3704900051048496494357433485559759053417848980008} a^{10} + \frac{153381701491545293486018572722243634262715185}{3704900051048496494357433485559759053417848980008} a^{9} + \frac{374332809369974952592514135441772927493985529}{1852450025524248247178716742779879526708924490004} a^{8} - \frac{1366060047144866495611739311784769800271418341}{1852450025524248247178716742779879526708924490004} a^{7} - \frac{8563765070404535498694382182862780476617488377}{1852450025524248247178716742779879526708924490004} a^{6} + \frac{6182081411122690569906046689000499842170240847}{926225012762124123589358371389939763354462245002} a^{5} + \frac{44222360884628622693898375377597241089048142899}{926225012762124123589358371389939763354462245002} a^{4} + \frac{64748870609144272927953836815096630500009460035}{463112506381062061794679185694969881677231122501} a^{3} + \frac{90061499692938985118582693634185290819694492571}{132317858966017731941336910198562823336351749286} a^{2} + \frac{32281555014435528966666068012940187922068851147}{9451275640429837995809779299897344524025124949} a + \frac{10562248531641044403599629950812721191908892224}{1350182234347119713687111328556763503432160707} \) (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:  \( 32762378537.631496 \) (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

\(\Q(\sqrt{-3}) \), 3.1.59300748.7 x3, 3.3.4941729.1, 6.0.10549736140078512.3, 6.0.172920981168.3 x2, 6.0.73262056528323.1, 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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
2Data not computed
3Data not computed
$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}$
$19$19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$