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Properties

Label 18.0.46649050280...4143.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{24}\cdot 13^{15}\cdot 19^{9}$
Root discriminant $159.89$
Ramified primes $3, 13, 19$
Class number $6368544$ (GRH)
Class group $[6, 108, 9828]$ (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![31296928456, -7873833456, 19161182850, -3879890761, 5007118512, -651424590, 669977697, -64188012, 64492053, -6413124, 5556303, -776070, 345457, -30510, 10839, -490, 165, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 165*x^16 - 490*x^15 + 10839*x^14 - 30510*x^13 + 345457*x^12 - 776070*x^11 + 5556303*x^10 - 6413124*x^9 + 64492053*x^8 - 64188012*x^7 + 669977697*x^6 - 651424590*x^5 + 5007118512*x^4 - 3879890761*x^3 + 19161182850*x^2 - 7873833456*x + 31296928456)
 
gp: K = bnfinit(x^18 - 3*x^17 + 165*x^16 - 490*x^15 + 10839*x^14 - 30510*x^13 + 345457*x^12 - 776070*x^11 + 5556303*x^10 - 6413124*x^9 + 64492053*x^8 - 64188012*x^7 + 669977697*x^6 - 651424590*x^5 + 5007118512*x^4 - 3879890761*x^3 + 19161182850*x^2 - 7873833456*x + 31296928456, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 165 x^{16} - 490 x^{15} + 10839 x^{14} - 30510 x^{13} + 345457 x^{12} - 776070 x^{11} + 5556303 x^{10} - 6413124 x^{9} + 64492053 x^{8} - 64188012 x^{7} + 669977697 x^{6} - 651424590 x^{5} + 5007118512 x^{4} - 3879890761 x^{3} + 19161182850 x^{2} - 7873833456 x + 31296928456 \)

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:  \(-4664905028044379350971453203047862914143=-\,3^{24}\cdot 13^{15}\cdot 19^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $159.89$
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$$\chi_{2223}(1,·)$, $\chi_{2223}(322,·)$, $\chi_{2223}(1291,·)$, $\chi_{2223}(1483,·)$, $\chi_{2223}(1804,·)$, $\chi_{2223}(1426,·)$, $\chi_{2223}(493,·)$, $\chi_{2223}(913,·)$, $\chi_{2223}(1234,·)$, $\chi_{2223}(550,·)$, $\chi_{2223}(742,·)$, $\chi_{2223}(1063,·)$, $\chi_{2223}(172,·)$, $\chi_{2223}(685,·)$, $\chi_{2223}(2032,·)$, $\chi_{2223}(1654,·)$, $\chi_{2223}(1975,·)$, $\chi_{2223}(2167,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{1667452} a^{15} - \frac{72279}{833726} a^{14} - \frac{362277}{1667452} a^{13} + \frac{156511}{833726} a^{12} - \frac{369109}{1667452} a^{11} - \frac{116957}{833726} a^{10} - \frac{167775}{1667452} a^{9} - \frac{99013}{416863} a^{8} + \frac{134065}{1667452} a^{7} + \frac{74590}{416863} a^{6} + \frac{580541}{1667452} a^{5} - \frac{9897}{416863} a^{4} - \frac{417259}{1667452} a^{3} + \frac{45491}{833726} a^{2} + \frac{175065}{833726} a + \frac{106560}{416863}$, $\frac{1}{1667452} a^{16} - \frac{35451}{1667452} a^{14} - \frac{15145}{416863} a^{13} - \frac{179757}{1667452} a^{12} - \frac{131231}{833726} a^{11} - \frac{48679}{1667452} a^{10} + \frac{77141}{416863} a^{9} - \frac{386531}{1667452} a^{8} - \frac{163983}{833726} a^{7} - \frac{241737}{1667452} a^{6} - \frac{54786}{416863} a^{5} - \frac{484099}{1667452} a^{4} - \frac{230309}{833726} a^{3} - \frac{167645}{833726} a^{2} - \frac{76477}{833726} a + \frac{178904}{416863}$, $\frac{1}{183979931150794592631550753581881331738038294554957385781359133463581564508} a^{17} + \frac{3325209192942159219658822197861733529617729993438553847312015685517}{91989965575397296315775376790940665869019147277478692890679566731790782254} a^{16} + \frac{25241111544880706897728957616029866872645512221798276970138786710315}{183979931150794592631550753581881331738038294554957385781359133463581564508} a^{15} - \frac{15907613253199310570293304803202209565211884840202684757302202216888176047}{183979931150794592631550753581881331738038294554957385781359133463581564508} a^{14} + \frac{38611948074195100000755428395066121829178089128152796234928094746318415259}{183979931150794592631550753581881331738038294554957385781359133463581564508} a^{13} - \frac{20782178134655124185440353183662536195597071030420111039914240790844773197}{183979931150794592631550753581881331738038294554957385781359133463581564508} a^{12} - \frac{41824048172935561693717333285893256058932012106938060650621767453029337377}{183979931150794592631550753581881331738038294554957385781359133463581564508} a^{11} + \frac{27662647186099629783084257535236324538003444049073990743098713710078736987}{183979931150794592631550753581881331738038294554957385781359133463581564508} a^{10} + \frac{2510985234923328903663216869834632774640232426581595787203450394942411373}{183979931150794592631550753581881331738038294554957385781359133463581564508} a^{9} - \frac{45590743895548192005918384191730557801024876532234334296613965381792685243}{183979931150794592631550753581881331738038294554957385781359133463581564508} a^{8} - \frac{39694645301512583037395901456149782395646772279851651858776654652653471655}{183979931150794592631550753581881331738038294554957385781359133463581564508} a^{7} + \frac{36445498743206359923329370506678355062479801307151239639466459012017927347}{183979931150794592631550753581881331738038294554957385781359133463581564508} a^{6} - \frac{89178816503605207235154507585248083690045709129836932748883792061626502513}{183979931150794592631550753581881331738038294554957385781359133463581564508} a^{5} - \frac{4631257443009712111373948106767271714594983217556486327457404221147904261}{183979931150794592631550753581881331738038294554957385781359133463581564508} a^{4} - \frac{15974622563713676438870331830090301188354550127627515133784726043834013081}{91989965575397296315775376790940665869019147277478692890679566731790782254} a^{3} + \frac{14759137034367689681339725622182579031419133052923597729011530006137483871}{183979931150794592631550753581881331738038294554957385781359133463581564508} a^{2} + \frac{13340047038964671611568485570006796499332777166700979105873592730767568239}{91989965575397296315775376790940665869019147277478692890679566731790782254} a - \frac{117860671594207035589130972051689180594018047450557257532321080573301607}{421972319153198606953098058674039751692748382006782994911374159320141203}$

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

Class group and class number

$C_{6}\times C_{108}\times C_{9828}$, which has order $6368544$ (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:  \( 400417.1364448253 \) (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

\(\Q(\sqrt{-247}) \), 3.3.169.1, 3.3.13689.1, \(\Q(\zeta_{9})^+\), 3.3.13689.2, 6.0.2546698687.1, 6.0.16708890085407.3, 6.0.98869172103.4, 6.0.16708890085407.4, 9.9.2565164201769.1

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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ 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 ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
3Data not computed
13Data not computed
$19$19.6.3.1$x^{6} - 38 x^{4} + 361 x^{2} - 109744$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
19.6.3.1$x^{6} - 38 x^{4} + 361 x^{2} - 109744$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
19.6.3.1$x^{6} - 38 x^{4} + 361 x^{2} - 109744$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$