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Properties

Label 18.0.22838497091...0368.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 3^{24}\cdot 61^{15}$
Root discriminant $376.25$
Ramified primes $2, 3, 61$
Class number $519710040$ (GRH)
Class group $[6, 114, 759810]$ (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![114567254733219, 3142777574502, 28764421042719, 292609182286, 3292755135573, 12508745586, 223808102424, 337218438, 9583959330, 4893800, 265060431, 9708, 4749545, -642, 53571, -6, 348, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 348*x^16 - 6*x^15 + 53571*x^14 - 642*x^13 + 4749545*x^12 + 9708*x^11 + 265060431*x^10 + 4893800*x^9 + 9583959330*x^8 + 337218438*x^7 + 223808102424*x^6 + 12508745586*x^5 + 3292755135573*x^4 + 292609182286*x^3 + 28764421042719*x^2 + 3142777574502*x + 114567254733219)
 
gp: K = bnfinit(x^18 + 348*x^16 - 6*x^15 + 53571*x^14 - 642*x^13 + 4749545*x^12 + 9708*x^11 + 265060431*x^10 + 4893800*x^9 + 9583959330*x^8 + 337218438*x^7 + 223808102424*x^6 + 12508745586*x^5 + 3292755135573*x^4 + 292609182286*x^3 + 28764421042719*x^2 + 3142777574502*x + 114567254733219, 1)
 

Normalized defining polynomial

\( x^{18} + 348 x^{16} - 6 x^{15} + 53571 x^{14} - 642 x^{13} + 4749545 x^{12} + 9708 x^{11} + 265060431 x^{10} + 4893800 x^{9} + 9583959330 x^{8} + 337218438 x^{7} + 223808102424 x^{6} + 12508745586 x^{5} + 3292755135573 x^{4} + 292609182286 x^{3} + 28764421042719 x^{2} + 3142777574502 x + 114567254733219 \)

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:  \(-22838497091409859087791143712903439974156730368=-\,2^{27}\cdot 3^{24}\cdot 61^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $376.25$
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:  $2, 3, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4392=2^{3}\cdot 3^{2}\cdot 61\)
Dirichlet character group:    $\lbrace$$\chi_{4392}(1,·)$, $\chi_{4392}(1219,·)$, $\chi_{4392}(1633,·)$, $\chi_{4392}(3979,·)$, $\chi_{4392}(2515,·)$, $\chi_{4392}(3097,·)$, $\chi_{4392}(3673,·)$, $\chi_{4392}(1051,·)$, $\chi_{4392}(2209,·)$, $\chi_{4392}(475,·)$, $\chi_{4392}(169,·)$, $\chi_{4392}(2929,·)$, $\chi_{4392}(4147,·)$, $\chi_{4392}(745,·)$, $\chi_{4392}(1465,·)$, $\chi_{4392}(3403,·)$, $\chi_{4392}(2683,·)$, $\chi_{4392}(1939,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{18} a^{9} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} + \frac{1}{6} a^{5} + \frac{1}{3} a^{4} + \frac{4}{9} a^{3} - \frac{1}{6} a^{2} - \frac{1}{2} a + \frac{1}{6}$, $\frac{1}{18} a^{10} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} + \frac{1}{6} a^{6} + \frac{1}{3} a^{5} + \frac{4}{9} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} + \frac{1}{6} a$, $\frac{1}{54} a^{11} + \frac{1}{54} a^{10} + \frac{1}{54} a^{9} + \frac{1}{18} a^{8} - \frac{1}{18} a^{7} - \frac{1}{18} a^{6} + \frac{4}{27} a^{5} - \frac{8}{27} a^{4} - \frac{7}{54} a^{3} - \frac{1}{6} a^{2} - \frac{4}{9} a + \frac{2}{9}$, $\frac{1}{972} a^{12} - \frac{2}{81} a^{10} - \frac{1}{243} a^{9} - \frac{17}{81} a^{8} + \frac{1}{27} a^{7} - \frac{4}{243} a^{6} - \frac{29}{81} a^{5} + \frac{53}{324} a^{4} + \frac{46}{243} a^{3} + \frac{5}{162} a^{2} + \frac{5}{27} a - \frac{55}{324}$, $\frac{1}{2916} a^{13} - \frac{1}{2916} a^{12} - \frac{2}{243} a^{11} + \frac{5}{729} a^{10} - \frac{19}{1458} a^{9} - \frac{41}{486} a^{8} + \frac{217}{1458} a^{7} + \frac{77}{1458} a^{6} + \frac{331}{972} a^{5} - \frac{461}{2916} a^{4} + \frac{85}{1458} a^{3} + \frac{25}{486} a^{2} + \frac{47}{972} a + \frac{217}{972}$, $\frac{1}{2916} a^{14} - \frac{1}{2916} a^{12} - \frac{1}{729} a^{11} + \frac{1}{54} a^{10} - \frac{14}{729} a^{9} + \frac{319}{1458} a^{8} - \frac{1}{486} a^{7} + \frac{277}{2916} a^{6} - \frac{11}{729} a^{5} + \frac{149}{972} a^{4} + \frac{505}{1458} a^{3} + \frac{13}{972} a^{2} - \frac{13}{162} a - \frac{293}{972}$, $\frac{1}{26244} a^{15} - \frac{5}{26244} a^{12} + \frac{16}{2187} a^{11} + \frac{25}{1458} a^{10} - \frac{89}{4374} a^{9} + \frac{101}{729} a^{8} - \frac{461}{2916} a^{7} + \frac{1513}{13122} a^{6} + \frac{103}{729} a^{5} + \frac{655}{2916} a^{4} - \frac{10807}{26244} a^{3} - \frac{1067}{4374} a^{2} - \frac{142}{729} a + \frac{1999}{8748}$, $\frac{1}{986876898090575384980436005764} a^{16} + \frac{5185729150091835173932085}{328958966030191794993478668588} a^{15} + \frac{143900073330081228785033}{36550996225576866110386518732} a^{14} - \frac{23614983654968342225619671}{986876898090575384980436005764} a^{13} - \frac{25274716345764790951000099}{54826494338365299165579778098} a^{12} + \frac{93371743965023697190967548}{27413247169182649582789889049} a^{11} + \frac{2996115796884683951736171943}{164479483015095897496739334294} a^{10} - \frac{615271287017158523202325577}{27413247169182649582789889049} a^{9} - \frac{22111156028731404260219430167}{109652988676730598331159556196} a^{8} + \frac{127256465670548736990209236373}{986876898090575384980436005764} a^{7} + \frac{46575160233486803445421963705}{328958966030191794993478668588} a^{6} + \frac{51213276579375535691158114369}{109652988676730598331159556196} a^{5} + \frac{325860804900659652527273136671}{986876898090575384980436005764} a^{4} - \frac{35458116331061650324121856569}{109652988676730598331159556196} a^{3} + \frac{8329273972245915925002824471}{36550996225576866110386518732} a^{2} + \frac{96641604635034829823927702893}{328958966030191794993478668588} a - \frac{262379505198079053758896186}{740898572140071610345672677}$, $\frac{1}{3904142580545522447901586256123651211311926884} a^{17} + \frac{3950567706269}{3904142580545522447901586256123651211311926884} a^{16} - \frac{6294703887002851883519511405922660981685}{1301380860181840815967195418707883737103975628} a^{15} - \frac{118229023333532162043258818716235308587127}{1952071290272761223950793128061825605655963442} a^{14} - \frac{354323277600304281175469782088403373573441}{3904142580545522447901586256123651211311926884} a^{13} + \frac{34367022799060166023613208388805976157507}{72298936676768934220399745483771318727998646} a^{12} + \frac{4985878121874099973145105491126604558705875}{650690430090920407983597709353941868551987814} a^{11} - \frac{511464670841990514441647528259373211505296}{325345215045460203991798854676970934275993907} a^{10} - \frac{10938002769513309899426742113067840853601417}{433793620060613605322398472902627912367991876} a^{9} - \frac{63824993080965593906033103408437606199561031}{3904142580545522447901586256123651211311926884} a^{8} + \frac{464675114277223545636964391011537604825822251}{3904142580545522447901586256123651211311926884} a^{7} + \frac{37519146423360883982361720692961129932365238}{325345215045460203991798854676970934275993907} a^{6} - \frac{27678608082099471270606250336462801961942791}{976035645136380611975396564030912802827981721} a^{5} - \frac{1324941462881630738927501401065247561924463105}{3904142580545522447901586256123651211311926884} a^{4} + \frac{55376983372124995112983676785561928020436171}{144597873353537868440799490967542637455997292} a^{3} + \frac{10514344836670889790235651821968360308996075}{325345215045460203991798854676970934275993907} a^{2} + \frac{268897285704363561636336203970378349810852859}{1301380860181840815967195418707883737103975628} a + \frac{2311958098304336305552164609221199094349483}{5862075946765048720572952336521998815783674}$

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

Class group and class number

$C_{6}\times C_{114}\times C_{759810}$, which has order $519710040$ (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:  \( 155231848.66582242 \) (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

\(\Q(\sqrt{-122}) \), \(\Q(\zeta_{9})^+\), 3.3.3721.1, 3.3.301401.1, 3.3.301401.2, 6.0.762481838592.7, 6.0.432433306112.1, 6.0.2837194921400832.2, 6.0.2837194921400832.1, 9.9.27380039270784201.1

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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
$3$3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
61Data not computed