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Properties

Label 18.0.38778658112...6643.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{33}\cdot 17^{8}$
Root discriminant $26.40$
Ramified primes $3, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3^2:S_3$ (as 18T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![576, -1728, 2160, -3000, 13104, -34002, 46231, -33267, 10344, 2630, -3168, 270, 780, -369, 9, 41, -6, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 6*x^16 + 41*x^15 + 9*x^14 - 369*x^13 + 780*x^12 + 270*x^11 - 3168*x^10 + 2630*x^9 + 10344*x^8 - 33267*x^7 + 46231*x^6 - 34002*x^5 + 13104*x^4 - 3000*x^3 + 2160*x^2 - 1728*x + 576)
 
gp: K = bnfinit(x^18 - 3*x^17 - 6*x^16 + 41*x^15 + 9*x^14 - 369*x^13 + 780*x^12 + 270*x^11 - 3168*x^10 + 2630*x^9 + 10344*x^8 - 33267*x^7 + 46231*x^6 - 34002*x^5 + 13104*x^4 - 3000*x^3 + 2160*x^2 - 1728*x + 576, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 6 x^{16} + 41 x^{15} + 9 x^{14} - 369 x^{13} + 780 x^{12} + 270 x^{11} - 3168 x^{10} + 2630 x^{9} + 10344 x^{8} - 33267 x^{7} + 46231 x^{6} - 34002 x^{5} + 13104 x^{4} - 3000 x^{3} + 2160 x^{2} - 1728 x + 576 \)

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:  \(-38778658112119365306896643=-\,3^{33}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.40$
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:  $3, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{6} + \frac{1}{3} a^{3}$, $\frac{1}{6} a^{10} - \frac{1}{3} a^{7} - \frac{1}{2} a^{6} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{11} - \frac{1}{3} a^{8} - \frac{1}{2} a^{7} - \frac{1}{3} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{6} a^{3}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{9} - \frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{6} a^{4} + \frac{1}{3} a^{3}$, $\frac{1}{36} a^{14} - \frac{1}{12} a^{13} - \frac{1}{12} a^{11} - \frac{1}{12} a^{10} + \frac{1}{12} a^{9} - \frac{1}{6} a^{8} + \frac{1}{6} a^{7} + \frac{1}{3} a^{6} - \frac{4}{9} a^{5} - \frac{1}{2} a^{4} + \frac{1}{12} a^{3} + \frac{5}{12} a^{2} - \frac{1}{2} a$, $\frac{1}{936} a^{15} + \frac{1}{104} a^{14} - \frac{1}{12} a^{13} - \frac{25}{312} a^{12} + \frac{1}{104} a^{11} - \frac{7}{312} a^{10} - \frac{1}{78} a^{9} - \frac{19}{52} a^{8} + \frac{23}{78} a^{7} - \frac{155}{468} a^{6} + \frac{6}{13} a^{5} - \frac{125}{312} a^{4} - \frac{29}{104} a^{3} - \frac{21}{52} a^{2} - \frac{5}{13} a + \frac{2}{13}$, $\frac{1}{1872} a^{16} - \frac{1}{1872} a^{15} - \frac{1}{156} a^{14} - \frac{25}{624} a^{13} - \frac{7}{624} a^{12} - \frac{37}{624} a^{11} + \frac{7}{312} a^{10} - \frac{11}{312} a^{9} - \frac{43}{156} a^{8} + \frac{259}{936} a^{7} + \frac{103}{468} a^{6} - \frac{19}{208} a^{5} - \frac{63}{208} a^{4} + \frac{23}{52} a^{3} - \frac{9}{52} a^{2} - \frac{1}{2} a + \frac{3}{13}$, $\frac{1}{7475463300219892224807456} a^{17} + \frac{293615788987124145917}{7475463300219892224807456} a^{16} + \frac{15556445918491022959}{95839273079742208010352} a^{15} - \frac{7391803494825531249053}{2491821100073297408269152} a^{14} - \frac{176612997811497117726149}{2491821100073297408269152} a^{13} + \frac{9810388505697471336383}{830607033357765802756384} a^{12} - \frac{23915711017865234154995}{622955275018324352067288} a^{11} - \frac{28712098392553175481097}{415303516678882901378192} a^{10} - \frac{36850477149086598589249}{311477637509162176033644} a^{9} + \frac{335120742344868644513467}{3737731650109946112403728} a^{8} + \frac{194333721577445602623041}{934432912527486528100932} a^{7} + \frac{507797713627642918472975}{2491821100073297408269152} a^{6} + \frac{25785155019172898879039}{830607033357765802756384} a^{5} - \frac{45482428373223139850411}{95839273079742208010352} a^{4} + \frac{18522002520679602847027}{155738818754581088016822} a^{3} - \frac{7758909791644102137429}{25956469792430181336137} a^{2} + \frac{355669480249215988288}{25956469792430181336137} a + \frac{4880887101954782676866}{25956469792430181336137}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( -\frac{8641960747811}{5681155212053256} a^{17} + \frac{14084439966657}{3787436808035504} a^{16} + \frac{126245783610055}{11362310424106512} a^{15} - \frac{319163787240127}{5681155212053256} a^{14} - \frac{165822075190931}{3787436808035504} a^{13} + \frac{6083601799943125}{11362310424106512} a^{12} - \frac{3405238504289649}{3787436808035504} a^{11} - \frac{1680198942109893}{1893718404017752} a^{10} + \frac{8239144065445373}{1893718404017752} a^{9} - \frac{1235679016409417}{710144401506657} a^{8} - \frac{31460327080354911}{1893718404017752} a^{7} + \frac{18287582890119199}{437011939388712} a^{6} - \frac{546452586169214965}{11362310424106512} a^{5} + \frac{94645620481335155}{3787436808035504} a^{4} - \frac{5033820508526183}{1420288803013314} a^{3} - \frac{1321759593072769}{946859202008876} a^{2} - \frac{966297020073063}{473429601004438} a + \frac{439584875409588}{236714800502219} \) (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:  \( 1172990.2101607663 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^2:S_3$ (as 18T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 10 conjugacy class representatives for $C_3^2:S_3$
Character table for $C_3^2:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.243.1 x3, 6.0.177147.2, 9.3.1198435061547.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 9 siblings: data not computed
Degree 18 siblings: 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$3$3.6.11.9$x^{6} + 3$$6$$1$$11$$S_3$$[5/2]_{2}$
3.6.11.9$x^{6} + 3$$6$$1$$11$$S_3$$[5/2]_{2}$
3.6.11.9$x^{6} + 3$$6$$1$$11$$S_3$$[5/2]_{2}$
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$