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Properties

Label 18.0.39231415897...8208.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{9}\cdot 17^{16}$
Root discriminant $34.12$
Ramified primes $2, 3, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_9$ (as 18T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![972, 1944, 648, -1296, -3132, 4374, 15885, 7038, 1671, -912, -1934, 918, 239, -456, 198, -66, 23, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 23*x^16 - 66*x^15 + 198*x^14 - 456*x^13 + 239*x^12 + 918*x^11 - 1934*x^10 - 912*x^9 + 1671*x^8 + 7038*x^7 + 15885*x^6 + 4374*x^5 - 3132*x^4 - 1296*x^3 + 648*x^2 + 1944*x + 972)
 
gp: K = bnfinit(x^18 - 6*x^17 + 23*x^16 - 66*x^15 + 198*x^14 - 456*x^13 + 239*x^12 + 918*x^11 - 1934*x^10 - 912*x^9 + 1671*x^8 + 7038*x^7 + 15885*x^6 + 4374*x^5 - 3132*x^4 - 1296*x^3 + 648*x^2 + 1944*x + 972, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 23 x^{16} - 66 x^{15} + 198 x^{14} - 456 x^{13} + 239 x^{12} + 918 x^{11} - 1934 x^{10} - 912 x^{9} + 1671 x^{8} + 7038 x^{7} + 15885 x^{6} + 4374 x^{5} - 3132 x^{4} - 1296 x^{3} + 648 x^{2} + 1944 x + 972 \)

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:  \(-3923141589765123982587998208=-\,2^{12}\cdot 3^{9}\cdot 17^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.12$
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:  $2, 3, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{2}$, $\frac{1}{18} a^{9} - \frac{1}{6} a^{8} - \frac{1}{6} a^{6} + \frac{1}{3} a^{5} - \frac{1}{6} a^{4} + \frac{5}{18} a^{3} + \frac{1}{3} a$, $\frac{1}{18} a^{10} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{6} a^{5} - \frac{2}{9} a^{4} - \frac{1}{6} a^{3}$, $\frac{1}{54} a^{11} + \frac{1}{54} a^{10} - \frac{1}{6} a^{8} + \frac{1}{18} a^{6} - \frac{1}{54} a^{5} - \frac{8}{27} a^{4} + \frac{1}{3} a^{3} + \frac{2}{9} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{540} a^{12} + \frac{1}{135} a^{11} - \frac{1}{45} a^{10} - \frac{1}{90} a^{9} + \frac{7}{60} a^{8} - \frac{1}{90} a^{7} + \frac{2}{135} a^{6} - \frac{113}{270} a^{5} + \frac{31}{180} a^{4} + \frac{13}{30} a^{3} - \frac{2}{15} a^{2} - \frac{7}{30} a - \frac{1}{5}$, $\frac{1}{1620} a^{13} + \frac{1}{1620} a^{12} + \frac{1}{270} a^{11} + \frac{1}{54} a^{10} - \frac{1}{180} a^{9} + \frac{17}{108} a^{8} - \frac{61}{405} a^{7} - \frac{8}{81} a^{6} - \frac{203}{540} a^{5} - \frac{11}{36} a^{4} - \frac{23}{90} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{7}{15}$, $\frac{1}{4860} a^{14} + \frac{1}{2430} a^{12} + \frac{1}{405} a^{11} - \frac{31}{1620} a^{10} + \frac{17}{810} a^{9} + \frac{61}{2430} a^{8} - \frac{14}{405} a^{7} + \frac{337}{4860} a^{6} - \frac{121}{270} a^{5} + \frac{13}{90} a^{4} + \frac{1}{5} a^{3} - \frac{13}{45} a^{2} + \frac{1}{5} a + \frac{2}{9}$, $\frac{1}{4860} a^{15} - \frac{1}{4860} a^{13} + \frac{11}{1620} a^{11} + \frac{1}{162} a^{10} - \frac{67}{4860} a^{9} + \frac{2}{81} a^{8} + \frac{313}{4860} a^{7} + \frac{13}{162} a^{6} + \frac{217}{540} a^{5} + \frac{7}{54} a^{4} - \frac{11}{45} a^{3} - \frac{1}{18} a^{2} - \frac{2}{45} a + \frac{1}{3}$, $\frac{1}{160380} a^{16} + \frac{1}{26730} a^{15} + \frac{7}{80190} a^{14} + \frac{1}{13365} a^{13} - \frac{1}{1620} a^{12} + \frac{191}{26730} a^{11} + \frac{1097}{40095} a^{10} + \frac{293}{13365} a^{9} + \frac{17173}{160380} a^{8} + \frac{13}{165} a^{7} - \frac{661}{13365} a^{6} + \frac{959}{2970} a^{5} + \frac{283}{2970} a^{4} + \frac{13}{33} a^{3} + \frac{628}{1485} a^{2} - \frac{6}{55} a - \frac{112}{495}$, $\frac{1}{97668493435638180} a^{17} + \frac{50350799017}{32556164478546060} a^{16} + \frac{2190386789759}{97668493435638180} a^{15} + \frac{134745436219}{1627808223927303} a^{14} - \frac{2509288457}{33254509171140} a^{13} + \frac{2285087077237}{32556164478546060} a^{12} + \frac{22805468746975}{19533698687127636} a^{11} - \frac{28966696923851}{2713013706545505} a^{10} - \frac{488590004391823}{19533698687127636} a^{9} + \frac{1893952440244193}{32556164478546060} a^{8} + \frac{977134232384549}{6511232895709212} a^{7} + \frac{861452243932}{100481989131315} a^{6} - \frac{523010431191779}{1808675804363670} a^{5} + \frac{8042121484922}{301445967393945} a^{4} + \frac{7337985428941}{20322200049030} a^{3} - \frac{16520021191519}{301445967393945} a^{2} - \frac{148845830873984}{301445967393945} a + \frac{714334613908}{3044908761555}$

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:  \( -\frac{211419137149}{328850146247940} a^{17} + \frac{1438517377063}{328850146247940} a^{16} - \frac{6021606467227}{328850146247940} a^{15} + \frac{6250608683869}{109616715415980} a^{14} - \frac{53034297044}{307912121955} a^{13} + \frac{11727561629786}{27404178853995} a^{12} - \frac{32028094650781}{65770029249588} a^{11} - \frac{76317581181199}{328850146247940} a^{10} + \frac{49779234595423}{32885014624794} a^{9} - \frac{19001173739117}{27404178853995} a^{8} - \frac{4742537215001}{7307781027732} a^{7} - \frac{132874902031291}{36538905138660} a^{6} - \frac{88174025921363}{12179635046220} a^{5} + \frac{34010812137479}{12179635046220} a^{4} - \frac{35786006111}{22808305330} a^{3} - \frac{1535157599383}{2029939174370} a^{2} - \frac{389442723279}{2029939174370} a - \frac{246770994634}{1014969587185} \) (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:  \( 352841796.026 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_9$ (as 18T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 6 conjugacy class representatives for $D_9$
Character table for $D_9$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.867.1 x3, 6.0.2255067.2, 9.1.36162326574144.2 x9

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

Sibling fields

Degree 9 sibling: 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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$2$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$3$3.2.1.2$x^{2} + 3$$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}$
3.2.1.2$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$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$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}$
17Data not computed