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Properties

Label 18.0.21946456682...4983.4
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{24}\cdot 13^{15}\cdot 19^{15}$
Root discriminant $426.65$
Ramified primes $3, 13, 19$
Class number $3906734328$ (GRH)
Class group $[21, 42, 378, 11718]$ (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![22504454258688, -618217616304, 5972918763024, 45178635864, 751115946456, -16181516019, 64122556561, -459373590, 3774087570, 69650435, 133354239, 4265028, 2403572, 58947, 9855, -550, -126, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 126*x^16 - 550*x^15 + 9855*x^14 + 58947*x^13 + 2403572*x^12 + 4265028*x^11 + 133354239*x^10 + 69650435*x^9 + 3774087570*x^8 - 459373590*x^7 + 64122556561*x^6 - 16181516019*x^5 + 751115946456*x^4 + 45178635864*x^3 + 5972918763024*x^2 - 618217616304*x + 22504454258688)
 
gp: K = bnfinit(x^18 - 3*x^17 - 126*x^16 - 550*x^15 + 9855*x^14 + 58947*x^13 + 2403572*x^12 + 4265028*x^11 + 133354239*x^10 + 69650435*x^9 + 3774087570*x^8 - 459373590*x^7 + 64122556561*x^6 - 16181516019*x^5 + 751115946456*x^4 + 45178635864*x^3 + 5972918763024*x^2 - 618217616304*x + 22504454258688, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 126 x^{16} - 550 x^{15} + 9855 x^{14} + 58947 x^{13} + 2403572 x^{12} + 4265028 x^{11} + 133354239 x^{10} + 69650435 x^{9} + 3774087570 x^{8} - 459373590 x^{7} + 64122556561 x^{6} - 16181516019 x^{5} + 751115946456 x^{4} + 45178635864 x^{3} + 5972918763024 x^{2} - 618217616304 x + 22504454258688 \)

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:  \(-219464566825677533664660221787658595963084794983=-\,3^{24}\cdot 13^{15}\cdot 19^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $426.65$
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:  $3, 13, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2223=3^{2}\cdot 13\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{2223}(1,·)$, $\chi_{2223}(322,·)$, $\chi_{2223}(259,·)$, $\chi_{2223}(1414,·)$, $\chi_{2223}(391,·)$, $\chi_{2223}(1291,·)$, $\chi_{2223}(1234,·)$, $\chi_{2223}(919,·)$, $\chi_{2223}(1816,·)$, $\chi_{2223}(160,·)$, $\chi_{2223}(1570,·)$, $\chi_{2223}(103,·)$, $\chi_{2223}(316,·)$, $\chi_{2223}(1426,·)$, $\chi_{2223}(1717,·)$, $\chi_{2223}(1654,·)$, $\chi_{2223}(1147,·)$, $\chi_{2223}(2044,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{3}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{64} a^{7} + \frac{1}{32} a^{5} + \frac{1}{64} a^{3} - \frac{1}{16} a$, $\frac{1}{128} a^{8} + \frac{1}{64} a^{6} + \frac{1}{128} a^{4} - \frac{1}{32} a^{2}$, $\frac{1}{256} a^{9} - \frac{1}{256} a^{8} - \frac{1}{128} a^{6} - \frac{3}{256} a^{5} - \frac{1}{256} a^{4} - \frac{3}{128} a^{3} + \frac{1}{64} a^{2} + \frac{1}{32} a$, $\frac{1}{512} a^{10} - \frac{1}{512} a^{9} - \frac{1}{256} a^{7} + \frac{13}{512} a^{6} + \frac{15}{512} a^{5} - \frac{11}{256} a^{4} - \frac{19}{128} a^{3} - \frac{15}{64} a^{2} - \frac{1}{8} a$, $\frac{1}{3072} a^{11} + \frac{1}{1536} a^{10} - \frac{5}{3072} a^{9} - \frac{3}{1024} a^{7} + \frac{15}{512} a^{6} - \frac{35}{3072} a^{5} + \frac{5}{768} a^{4} + \frac{7}{48} a^{3} + \frac{3}{64} a^{2} - \frac{3}{64} a$, $\frac{1}{147456} a^{12} + \frac{7}{49152} a^{11} + \frac{31}{49152} a^{10} + \frac{193}{147456} a^{9} - \frac{61}{16384} a^{8} - \frac{355}{49152} a^{7} - \frac{4193}{147456} a^{6} + \frac{2425}{49152} a^{5} - \frac{163}{6144} a^{4} + \frac{4391}{18432} a^{3} - \frac{35}{3072} a^{2} + \frac{323}{1024} a - \frac{1}{3}$, $\frac{1}{294912} a^{13} + \frac{1}{8192} a^{11} + \frac{5}{9216} a^{10} + \frac{65}{49152} a^{9} + \frac{1}{3072} a^{8} + \frac{223}{73728} a^{7} - \frac{17}{1024} a^{6} - \frac{1751}{32768} a^{5} + \frac{485}{9216} a^{4} + \frac{937}{12288} a^{3} - \frac{45}{256} a^{2} - \frac{1021}{6144} a$, $\frac{1}{7077888} a^{14} + \frac{1}{786432} a^{13} - \frac{1}{589824} a^{12} + \frac{61}{1769472} a^{11} - \frac{317}{393216} a^{10} - \frac{1967}{1179648} a^{9} + \frac{4147}{1769472} a^{8} - \frac{989}{196608} a^{7} - \frac{18901}{2359296} a^{6} - \frac{255863}{7077888} a^{5} + \frac{17429}{294912} a^{4} + \frac{1161}{32768} a^{3} + \frac{4187}{147456} a^{2} - \frac{6061}{16384} a - \frac{1}{8}$, $\frac{1}{28311552} a^{15} + \frac{1}{9437184} a^{13} - \frac{1}{884736} a^{12} + \frac{73}{1572864} a^{11} + \frac{25}{36864} a^{10} - \frac{5317}{14155776} a^{9} - \frac{35}{49152} a^{8} + \frac{45415}{9437184} a^{7} - \frac{3793}{221184} a^{6} + \frac{333229}{9437184} a^{5} + \frac{12421}{294912} a^{4} - \frac{86081}{393216} a^{3} - \frac{1267}{24576} a^{2} - \frac{62177}{196608} a + \frac{19}{96}$, $\frac{1}{452984832} a^{16} - \frac{1}{150994944} a^{15} + \frac{19}{452984832} a^{14} + \frac{295}{452984832} a^{13} - \frac{13}{8388608} a^{12} + \frac{11309}{226492416} a^{11} + \frac{214955}{226492416} a^{10} - \frac{48793}{25165824} a^{9} - \frac{1561931}{452984832} a^{8} + \frac{862753}{452984832} a^{7} - \frac{1094819}{150994944} a^{6} - \frac{12019301}{452984832} a^{5} + \frac{1076689}{18874368} a^{4} + \frac{3583753}{18874368} a^{3} - \frac{1827355}{9437184} a^{2} + \frac{514675}{3145728} a - \frac{553}{1536}$, $\frac{1}{1590473695398964773859101392502793662931956238052448666648576} a^{17} - \frac{274539347103862776527791575512068428735626142531965}{795236847699482386929550696251396831465978119026224333324288} a^{16} - \frac{584089231652706539661285109581784662922871283256753}{198809211924870596732387674062849207866494529756556083331072} a^{15} - \frac{54260310163975005734694177090325824767827877553335191}{795236847699482386929550696251396831465978119026224333324288} a^{14} + \frac{526343483621256187822121667460667869846125521715501857}{1590473695398964773859101392502793662931956238052448666648576} a^{13} - \frac{1211718715396846764415256146540764293356581213251212181}{397618423849741193464775348125698415732989059513112166662144} a^{12} + \frac{72343179618470110853995716441477807053151871007191343}{12425575745304412295774229628928075491655908109784755208192} a^{11} - \frac{158388088888910885069572005246831501621809968983205410223}{397618423849741193464775348125698415732989059513112166662144} a^{10} + \frac{876563930548216057623583554036425239116919930523257644259}{1590473695398964773859101392502793662931956238052448666648576} a^{9} + \frac{1799092047545536703138015680346177017594526919224007864767}{795236847699482386929550696251396831465978119026224333324288} a^{8} - \frac{185824536760502764676667929914885271979164288251472151961}{24851151490608824591548459257856150983311816219569510416384} a^{7} - \frac{22193235736903218460532226864338711418307729656536562122827}{795236847699482386929550696251396831465978119026224333324288} a^{6} - \frac{9492179772675545764539640232374289674155089447228676732941}{176719299488773863762122376944754851436884026450272074072064} a^{5} + \frac{1552403586729389062800908674263262992796182180741405587249}{33134868654145099455397945677141534644415754959426013888512} a^{4} - \frac{586189971315058707112349207551834262857094024569850667109}{66269737308290198910795891354283069288831509918852027777024} a^{3} + \frac{883181445622495456419916619155483449469568081201439605137}{5522478109024183242566324279523589107402625826571002314752} a^{2} - \frac{3667358345244766100781230346621428066833483912583618308213}{11044956218048366485132648559047178214805251653142004629504} a - \frac{665443576635149791314657498724751791526929687045522417}{5393045028343928947818676054222254987697876783760744448}$

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

Class group and class number

$C_{21}\times C_{42}\times C_{378}\times C_{11718}$, which has order $3906734328$ (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:  \( 931139254937.0632 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{-247}) \), 3.3.13689.1, 3.3.29241.1, 3.3.61009.1, 3.3.4941729.3, 6.0.16708890085407.3, 6.0.35691771129183.4, 6.0.919358226007.2, 6.0.6031909320831927.2, 9.9.120680409781884363489.9

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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{18}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$3$3.6.8.4$x^{6} + 18 x^{2} + 63$$3$$2$$8$$C_6$$[2]^{2}$
3.6.8.4$x^{6} + 18 x^{2} + 63$$3$$2$$8$$C_6$$[2]^{2}$
3.6.8.4$x^{6} + 18 x^{2} + 63$$3$$2$$8$$C_6$$[2]^{2}$
$13$13.6.5.4$x^{6} + 26$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.4$x^{6} + 26$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.4$x^{6} + 26$$6$$1$$5$$C_6$$[\ ]_{6}$
19Data not computed