Properties

Label 18.0.16481672572...4375.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,5^{4}\cdot 11^{4}\cdot 23^{9}$
Root discriminant $11.68$
Ramified primes $5, 11, 23$
Class number $1$
Class group Trivial
Galois Group $C_3\wr C_3:C_2$ (as 18T88)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, 14, -37, 52, -57, 46, -17, -8, 19, -11, 10, -5, 0, -5, -1, 2, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 2*x^16 - x^15 - 5*x^14 - 5*x^12 + 10*x^11 - 11*x^10 + 19*x^9 - 8*x^8 - 17*x^7 + 46*x^6 - 57*x^5 + 52*x^4 - 37*x^3 + 14*x^2 - 3*x + 1)
gp: K = bnfinit(x^18 + 2*x^16 - x^15 - 5*x^14 - 5*x^12 + 10*x^11 - 11*x^10 + 19*x^9 - 8*x^8 - 17*x^7 + 46*x^6 - 57*x^5 + 52*x^4 - 37*x^3 + 14*x^2 - 3*x + 1, 1)

Normalized defining polynomial

\(x^{18} \) \(\mathstrut +\mathstrut 2 x^{16} \) \(\mathstrut -\mathstrut x^{15} \) \(\mathstrut -\mathstrut 5 x^{14} \) \(\mathstrut -\mathstrut 5 x^{12} \) \(\mathstrut +\mathstrut 10 x^{11} \) \(\mathstrut -\mathstrut 11 x^{10} \) \(\mathstrut +\mathstrut 19 x^{9} \) \(\mathstrut -\mathstrut 8 x^{8} \) \(\mathstrut -\mathstrut 17 x^{7} \) \(\mathstrut +\mathstrut 46 x^{6} \) \(\mathstrut -\mathstrut 57 x^{5} \) \(\mathstrut +\mathstrut 52 x^{4} \) \(\mathstrut -\mathstrut 37 x^{3} \) \(\mathstrut +\mathstrut 14 x^{2} \) \(\mathstrut -\mathstrut 3 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(-16481672572799864375=-\,5^{4}\cdot 11^{4}\cdot 23^{9}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.68$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $5, 11, 23$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{5} a^{15} + \frac{1}{5} a^{14} + \frac{1}{5} a^{13} + \frac{1}{5} a^{12} + \frac{2}{5} a^{11} - \frac{2}{5} a^{10} + \frac{2}{5} a^{9} + \frac{2}{5} a^{8} + \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} - \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{16} + \frac{1}{5} a^{12} + \frac{1}{5} a^{11} - \frac{1}{5} a^{10} - \frac{1}{5} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{2}{5}$, $\frac{1}{3029771545} a^{17} - \frac{292269546}{3029771545} a^{16} + \frac{113494807}{3029771545} a^{15} - \frac{676258248}{3029771545} a^{14} - \frac{624457397}{3029771545} a^{13} - \frac{1001474408}{3029771545} a^{12} - \frac{1233781428}{3029771545} a^{11} + \frac{701788097}{3029771545} a^{10} - \frac{217782827}{3029771545} a^{9} - \frac{804115889}{3029771545} a^{8} + \frac{257956867}{3029771545} a^{7} - \frac{138295421}{605954309} a^{6} + \frac{160462277}{605954309} a^{5} + \frac{490679776}{3029771545} a^{4} + \frac{23624492}{605954309} a^{3} - \frac{1190301754}{3029771545} a^{2} + \frac{1455677013}{3029771545} a + \frac{1387349104}{3029771545}$

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

Class group and class number

Trivial Abelian group, order $1$

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:  \( \frac{1970603636}{3029771545} a^{17} + \frac{2188112856}{3029771545} a^{16} + \frac{5530995419}{3029771545} a^{15} + \frac{3399328134}{3029771545} a^{14} - \frac{1624862711}{605954309} a^{13} - \frac{9996481564}{3029771545} a^{12} - \frac{16986702342}{3029771545} a^{11} + \frac{4244500771}{3029771545} a^{10} - \frac{11383889953}{3029771545} a^{9} + \frac{21155039043}{3029771545} a^{8} + \frac{12489846706}{3029771545} a^{7} - \frac{28224723489}{3029771545} a^{6} + \frac{11064414150}{605954309} a^{5} - \frac{35883170422}{3029771545} a^{4} + \frac{36276346071}{3029771545} a^{3} - \frac{16368322243}{3029771545} a^{2} - \frac{4338763561}{3029771545} a - \frac{4133725821}{3029771545} \),  \( \frac{1479144098}{3029771545} a^{17} + \frac{1522240673}{3029771545} a^{16} + \frac{3956095197}{3029771545} a^{15} + \frac{2206770102}{3029771545} a^{14} - \frac{1315669487}{605954309} a^{13} - \frac{7394147687}{3029771545} a^{12} - \frac{12609467876}{3029771545} a^{11} + \frac{3322782878}{3029771545} a^{10} - \frac{8287992694}{3029771545} a^{9} + \frac{18219348324}{3029771545} a^{8} + \frac{11971924238}{3029771545} a^{7} - \frac{21299829087}{3029771545} a^{6} + \frac{8603570360}{605954309} a^{5} - \frac{30284280881}{3029771545} a^{4} + \frac{23658661313}{3029771545} a^{3} - \frac{9454765954}{3029771545} a^{2} - \frac{3560487678}{3029771545} a - \frac{2788157243}{3029771545} \),  \( \frac{24336452}{605954309} a^{17} + \frac{3298373}{605954309} a^{16} - \frac{8479036}{605954309} a^{15} - \frac{88164388}{605954309} a^{14} - \frac{323042460}{605954309} a^{13} - \frac{185736271}{605954309} a^{12} + \frac{31614020}{605954309} a^{11} + \frac{477455930}{605954309} a^{10} + \frac{457744720}{605954309} a^{9} + \frac{607510476}{605954309} a^{8} + \frac{403693276}{605954309} a^{7} - \frac{1072338463}{605954309} a^{6} + \frac{706054491}{605954309} a^{5} - \frac{411748723}{605954309} a^{4} - \frac{809334002}{605954309} a^{3} + \frac{590251126}{605954309} a^{2} - \frac{872556126}{605954309} a + \frac{64562806}{605954309} \),  \( \frac{2231209431}{3029771545} a^{17} + \frac{213070895}{605954309} a^{16} + \frac{5099750773}{3029771545} a^{15} + \frac{339307578}{3029771545} a^{14} - \frac{10812281816}{3029771545} a^{13} - \frac{4953283861}{3029771545} a^{12} - \frac{2864145163}{605954309} a^{11} + \frac{14952559549}{3029771545} a^{10} - \frac{3576565377}{605954309} a^{9} + \frac{33840404492}{3029771545} a^{8} - \frac{2850765336}{3029771545} a^{7} - \frac{39048644127}{3029771545} a^{6} + \frac{17490063547}{605954309} a^{5} - \frac{89167593903}{3029771545} a^{4} + \frac{78158916648}{3029771545} a^{3} - \frac{43711182016}{3029771545} a^{2} + \frac{6987847816}{3029771545} a - \frac{661590491}{3029771545} \),  \( \frac{2141844549}{3029771545} a^{17} + \frac{474171084}{605954309} a^{16} + \frac{6169538819}{3029771545} a^{15} + \frac{3850939419}{3029771545} a^{14} - \frac{8413245237}{3029771545} a^{13} - \frac{10345056262}{3029771545} a^{12} - \frac{18831684471}{3029771545} a^{11} + \frac{4735385992}{3029771545} a^{10} - \frac{12635953821}{3029771545} a^{9} + \frac{23499525277}{3029771545} a^{8} + \frac{11949484773}{3029771545} a^{7} - \frac{31054415754}{3029771545} a^{6} + \frac{59312621386}{3029771545} a^{5} - \frac{44614309672}{3029771545} a^{4} + \frac{44298016219}{3029771545} a^{3} - \frac{17286913153}{3029771545} a^{2} - \frac{446451261}{605954309} a - \frac{2611432288}{3029771545} \),  \( \frac{62798952}{3029771545} a^{17} + \frac{1519346228}{3029771545} a^{16} + \frac{317507940}{605954309} a^{15} + \frac{788819537}{605954309} a^{14} + \frac{1629911697}{3029771545} a^{13} - \frac{1335467913}{605954309} a^{12} - \frac{6864065139}{3029771545} a^{11} - \frac{10695378143}{3029771545} a^{10} + \frac{5731601328}{3029771545} a^{9} - \frac{6789354101}{3029771545} a^{8} + \frac{3312512846}{605954309} a^{7} + \frac{4382689582}{3029771545} a^{6} - \frac{21923343522}{3029771545} a^{5} + \frac{41750092592}{3029771545} a^{4} - \frac{32638152494}{3029771545} a^{3} + \frac{5651008404}{605954309} a^{2} - \frac{11211426441}{3029771545} a - \frac{3189513834}{3029771545} \),  \( \frac{2053140866}{3029771545} a^{17} + \frac{2051356828}{3029771545} a^{16} + \frac{6086776662}{3029771545} a^{15} + \frac{3405588862}{3029771545} a^{14} - \frac{7837855387}{3029771545} a^{13} - \frac{9410546054}{3029771545} a^{12} - \frac{20281477869}{3029771545} a^{11} + \frac{3564272443}{3029771545} a^{10} - \frac{14071925587}{3029771545} a^{9} + \frac{29569204437}{3029771545} a^{8} + \frac{10828300606}{3029771545} a^{7} - \frac{25415081141}{3029771545} a^{6} + \frac{62622551198}{3029771545} a^{5} - \frac{11777547045}{605954309} a^{4} + \frac{55067368568}{3029771545} a^{3} - \frac{29185639114}{3029771545} a^{2} + \frac{5934806303}{3029771545} a - \frac{6168377713}{3029771545} \),  \( \frac{2023547642}{3029771545} a^{17} + \frac{1687271703}{3029771545} a^{16} + \frac{1143763339}{605954309} a^{15} + \frac{505362088}{605954309} a^{14} - \frac{7573673053}{3029771545} a^{13} - \frac{1345506649}{605954309} a^{12} - \frac{16738283774}{3029771545} a^{11} + \frac{8036592142}{3029771545} a^{10} - \frac{16004106962}{3029771545} a^{9} + \frac{27395991984}{3029771545} a^{8} + \frac{241426321}{605954309} a^{7} - \frac{28001707463}{3029771545} a^{6} + \frac{65557409613}{3029771545} a^{5} - \frac{66903634313}{3029771545} a^{4} + \frac{67138368436}{3029771545} a^{3} - \frac{7987060061}{605954309} a^{2} + \frac{10779227694}{3029771545} a - \frac{4710336964}{3029771545} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 102.074554739 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_3\wr C_3:C_2$ (as 18T88):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 162
The 13 conjugacy class representatives for $C_3\wr C_3:C_2$
Character table for $C_3\wr C_3:C_2$

Intermediate fields

\(\Q(\sqrt{-23}) \), 3.1.23.1 x3, 6.0.12167.1, 9.1.846519025.1 x3

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$5$5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
5.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
$11$11.6.0.1$x^{6} + x^{2} - 2 x + 8$$1$$6$$0$$C_6$$[\ ]^{6}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
11.6.0.1$x^{6} + x^{2} - 2 x + 8$$1$$6$$0$$C_6$$[\ ]^{6}$
$23$23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.6.3.2$x^{6} - 529 x^{2} + 48668$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
23.6.3.2$x^{6} - 529 x^{2} + 48668$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$