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Properties

Label 18.0.40525551530...000.10
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{39}\cdot 5^{12}$
Root discriminant $50.17$
Ramified primes $2, 3, 5$
Class number $27$ (GRH)
Class group $[3, 3, 3]$ (GRH)
Galois group $C_3\times C_3:S_3$ (as 18T23)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![34896, -15120, 170424, -35748, -150804, -4428, 48987, 14490, 945, -4056, -3240, 270, 177, 72, 36, -18, 9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 9*x^16 - 18*x^15 + 36*x^14 + 72*x^13 + 177*x^12 + 270*x^11 - 3240*x^10 - 4056*x^9 + 945*x^8 + 14490*x^7 + 48987*x^6 - 4428*x^5 - 150804*x^4 - 35748*x^3 + 170424*x^2 - 15120*x + 34896)
 
gp: K = bnfinit(x^18 + 9*x^16 - 18*x^15 + 36*x^14 + 72*x^13 + 177*x^12 + 270*x^11 - 3240*x^10 - 4056*x^9 + 945*x^8 + 14490*x^7 + 48987*x^6 - 4428*x^5 - 150804*x^4 - 35748*x^3 + 170424*x^2 - 15120*x + 34896, 1)
 

Normalized defining polynomial

\( x^{18} + 9 x^{16} - 18 x^{15} + 36 x^{14} + 72 x^{13} + 177 x^{12} + 270 x^{11} - 3240 x^{10} - 4056 x^{9} + 945 x^{8} + 14490 x^{7} + 48987 x^{6} - 4428 x^{5} - 150804 x^{4} - 35748 x^{3} + 170424 x^{2} - 15120 x + 34896 \)

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:  \(-4052555153018976267000000000000=-\,2^{12}\cdot 3^{39}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.17$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8} + \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{9} - \frac{3}{8} a^{5} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{10} + \frac{1}{8} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{5032} a^{15} - \frac{99}{5032} a^{13} + \frac{3}{148} a^{12} + \frac{35}{5032} a^{11} - \frac{45}{1258} a^{10} - \frac{169}{5032} a^{9} - \frac{625}{2516} a^{8} - \frac{927}{5032} a^{7} - \frac{5}{629} a^{6} + \frac{123}{5032} a^{5} + \frac{331}{2516} a^{4} + \frac{41}{2516} a^{3} - \frac{195}{1258} a^{2} - \frac{278}{629} a + \frac{58}{629}$, $\frac{1}{5032} a^{16} - \frac{99}{5032} a^{14} + \frac{3}{148} a^{13} + \frac{35}{5032} a^{12} - \frac{45}{1258} a^{11} - \frac{169}{5032} a^{10} + \frac{1}{629} a^{9} + \frac{331}{5032} a^{8} - \frac{5}{629} a^{7} - \frac{1135}{5032} a^{6} + \frac{331}{2516} a^{5} - \frac{147}{629} a^{4} - \frac{1019}{2516} a^{3} + \frac{73}{1258} a^{2} + \frac{58}{629} a$, $\frac{1}{1884049707377067207018094411200056} a^{17} - \frac{8678832172519237195212475021}{942024853688533603509047205600028} a^{16} + \frac{1448226846863888742800241203}{1884049707377067207018094411200056} a^{15} + \frac{18877294829508640144000622633771}{942024853688533603509047205600028} a^{14} + \frac{9542451907189133711157828447041}{235506213422133400877261801400007} a^{13} + \frac{21206971991026396587198400997161}{235506213422133400877261801400007} a^{12} - \frac{90322336468143389693223570017681}{1884049707377067207018094411200056} a^{11} + \frac{36017065688560498187484134938587}{942024853688533603509047205600028} a^{10} + \frac{20192592788576560767281217705339}{942024853688533603509047205600028} a^{9} + \frac{98546500474635492272643684994901}{942024853688533603509047205600028} a^{8} + \frac{317209985751968180253493232025}{18291744731816186475903829234952} a^{7} - \frac{12680713384592001307741127075177}{471012426844266801754523602800014} a^{6} - \frac{529351625266596884191626006825809}{1884049707377067207018094411200056} a^{5} - \frac{60477972874541175163513403772121}{235506213422133400877261801400007} a^{4} - \frac{412864730189507720112283846208989}{942024853688533603509047205600028} a^{3} + \frac{1605187410682115036232453308017}{471012426844266801754523602800014} a^{2} - \frac{25899866767699259944849339261399}{235506213422133400877261801400007} a + \frac{62718912393162488494390974641814}{235506213422133400877261801400007}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}$, which has order $27$ (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:  \( -\frac{3400791522608286}{261368613242854966729} a^{17} - \frac{4904243309668713}{522737226485709933458} a^{16} - \frac{122527587910196367}{1045474452971419866916} a^{15} + \frac{35707428017561700}{261368613242854966729} a^{14} - \frac{289355022742569051}{1045474452971419866916} a^{13} - \frac{359540844395741481}{261368613242854966729} a^{12} - \frac{640455504924426750}{261368613242854966729} a^{11} - \frac{3017987756394015339}{522737226485709933458} a^{10} + \frac{42402793399776490741}{1045474452971419866916} a^{9} + \frac{21714495737706724113}{261368613242854966729} a^{8} + \frac{9079973348150306367}{261368613242854966729} a^{7} - \frac{72851609948838117327}{522737226485709933458} a^{6} - \frac{766620758303786882187}{1045474452971419866916} a^{5} - \frac{242784982351165066425}{522737226485709933458} a^{4} + \frac{1794822637989828768015}{1045474452971419866916} a^{3} + \frac{572616304425115178895}{522737226485709933458} a^{2} - \frac{466802025763186591407}{261368613242854966729} a + \frac{199267579090262160650}{261368613242854966729} \) (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:  \( 16318156.115223706 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_3:S_3$ (as 18T23):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 18 conjugacy class representatives for $C_3\times C_3:S_3$
Character table for $C_3\times C_3:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.243.1 x3, 6.0.1771470000.8, 6.0.1771470000.7, 6.0.196830000.3, 6.0.177147.2

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

Sibling fields

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 ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
3Data not computed
$5$5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$