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Properties

Label 18.0.15412566112...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{9}\cdot 5^{6}\cdot 13^{10}\cdot 31^{6}$
Root discriminant $61.41$
Ramified primes $2, 3, 5, 13, 31$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_3\times S_3^2$ (as 18T46)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1161, 30537, 285795, 210708, 49959, 172251, 188785, -52869, -55196, 30186, 15447, -11589, 641, 147, 321, -75, -11, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 11*x^16 - 75*x^15 + 321*x^14 + 147*x^13 + 641*x^12 - 11589*x^11 + 15447*x^10 + 30186*x^9 - 55196*x^8 - 52869*x^7 + 188785*x^6 + 172251*x^5 + 49959*x^4 + 210708*x^3 + 285795*x^2 + 30537*x + 1161)
 
gp: K = bnfinit(x^18 - 11*x^16 - 75*x^15 + 321*x^14 + 147*x^13 + 641*x^12 - 11589*x^11 + 15447*x^10 + 30186*x^9 - 55196*x^8 - 52869*x^7 + 188785*x^6 + 172251*x^5 + 49959*x^4 + 210708*x^3 + 285795*x^2 + 30537*x + 1161, 1)
 

Normalized defining polynomial

\( x^{18} - 11 x^{16} - 75 x^{15} + 321 x^{14} + 147 x^{13} + 641 x^{12} - 11589 x^{11} + 15447 x^{10} + 30186 x^{9} - 55196 x^{8} - 52869 x^{7} + 188785 x^{6} + 172251 x^{5} + 49959 x^{4} + 210708 x^{3} + 285795 x^{2} + 30537 x + 1161 \)

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:  \(-154125661129436703526043328000000=-\,2^{12}\cdot 3^{9}\cdot 5^{6}\cdot 13^{10}\cdot 31^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61.41$
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:  $2, 3, 5, 13, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{12} - \frac{1}{3} a^{8} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{15} + \frac{1}{9} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{10} + \frac{2}{9} a^{9} + \frac{1}{3} a^{8} + \frac{1}{9} a^{5} - \frac{1}{3} a^{4} + \frac{4}{9} a^{3} + \frac{1}{3} a$, $\frac{1}{34371} a^{16} - \frac{493}{11457} a^{15} + \frac{1264}{34371} a^{14} + \frac{3892}{11457} a^{13} - \frac{926}{11457} a^{12} + \frac{655}{11457} a^{11} + \frac{1244}{34371} a^{10} + \frac{4481}{11457} a^{9} + \frac{4853}{11457} a^{8} - \frac{184}{3819} a^{7} + \frac{2539}{34371} a^{6} + \frac{199}{11457} a^{5} - \frac{6329}{34371} a^{4} + \frac{2195}{11457} a^{3} - \frac{1432}{11457} a^{2} + \frac{1310}{3819} a - \frac{443}{3819}$, $\frac{1}{74271410279686778625726529564409418516627556389} a^{17} - \frac{173528514515443599785545538662724063921281}{74271410279686778625726529564409418516627556389} a^{16} - \frac{3605504330228657708761074161933813517855921095}{74271410279686778625726529564409418516627556389} a^{15} + \frac{8327009892920240724177217408316928438687616016}{74271410279686778625726529564409418516627556389} a^{14} + \frac{56379789584422483877401228398623799475560943}{434335732629747243425301342481926424073845359} a^{13} + \frac{79648011026560085504342949294692927408172260}{634798378458861355775440423627430927492543217} a^{12} - \frac{34082435439770897723667246108241922566640714113}{74271410279686778625726529564409418516627556389} a^{11} + \frac{1953210101245988966404135936999587480788444842}{5713185406129752201978963812646878347432888953} a^{10} - \frac{2805035514952975842844823273049130143302108233}{8252378919965197625080725507156602057403061821} a^{9} + \frac{10356582862367719078980393982000858484178857119}{24757136759895592875242176521469806172209185463} a^{8} + \frac{26567537169562698990242316786060494076254034625}{74271410279686778625726529564409418516627556389} a^{7} - \frac{8666544543808528460729539111165850084679959026}{74271410279686778625726529564409418516627556389} a^{6} + \frac{4141404588626095746831079840211268935922924598}{74271410279686778625726529564409418516627556389} a^{5} - \frac{5532881843298569330842512827770727233713959938}{74271410279686778625726529564409418516627556389} a^{4} + \frac{780832903874905886730222524089098084013825566}{2750792973321732541693575169052200685801020607} a^{3} + \frac{1946607024561674448014121010063648618197247510}{24757136759895592875242176521469806172209185463} a^{2} + \frac{1439757439008139942623210015994297956981256904}{8252378919965197625080725507156602057403061821} a - \frac{1333890114895433328055480936995762064709819176}{8252378919965197625080725507156602057403061821}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( \frac{225870134970968454935051636}{2162915254159580529129852396249} a^{17} - \frac{7541670641543877254081904}{720971751386526843043284132083} a^{16} - \frac{828065039489618787578115514}{720971751386526843043284132083} a^{15} - \frac{5555717911027175180535250948}{720971751386526843043284132083} a^{14} + \frac{74151435637955055953987208520}{2162915254159580529129852396249} a^{13} + \frac{8568277340235851870479924991}{720971751386526843043284132083} a^{12} + \frac{140192991892965865228173653782}{2162915254159580529129852396249} a^{11} - \frac{873809216803819927729283219102}{720971751386526843043284132083} a^{10} + \frac{3736777842193481179988644136204}{2162915254159580529129852396249} a^{9} + \frac{2165522315529573061859647606695}{720971751386526843043284132083} a^{8} - \frac{13474389197839878758634537164092}{2162915254159580529129852396249} a^{7} - \frac{3243902532402500866976094027176}{720971751386526843043284132083} a^{6} + \frac{14378795291836331488648832424738}{720971751386526843043284132083} a^{5} + \frac{11229897221990829367184343345585}{720971751386526843043284132083} a^{4} + \frac{8687011358732154152369969950582}{2162915254159580529129852396249} a^{3} + \frac{16241055410062159964150738150082}{720971751386526843043284132083} a^{2} + \frac{19561212094396052456404847971206}{720971751386526843043284132083} a + \frac{1433259002026806861146308757962}{720971751386526843043284132083} \) (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:  \( 1503257555.4600825 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3^2$ (as 18T46):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 27 conjugacy class representatives for $C_3\times S_3^2$
Character table for $C_3\times S_3^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.780.1, 6.0.4385043.1, 6.0.1825200.1

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

Sibling fields

Degree 12 sibling: 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 R R R ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
2Data not computed
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$5$5.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$13$13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.2.3$x^{3} - 52$$3$$1$$2$$C_3$$[\ ]_{3}$
13.6.3.2$x^{6} - 338 x^{2} + 13182$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.6.5.6$x^{6} + 416$$6$$1$$5$$C_6$$[\ ]_{6}$
$31$31.3.0.1$x^{3} - x + 9$$1$$3$$0$$C_3$$[\ ]^{3}$
31.3.2.3$x^{3} - 1519$$3$$1$$2$$C_3$$[\ ]_{3}$
31.6.4.2$x^{6} - 31 x^{3} + 11532$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
31.6.0.1$x^{6} - 2 x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$