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Properties

Label 18.0.72436308017...1552.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 3^{44}\cdot 19^{17}$
Root discriminant $669.20$
Ramified primes $2, 3, 19$
Class number $3601594368$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 2, 36, 781596]$ (GRH)
Galois group $C_{18}$ (as 18T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![163208757248, 0, 27472624091136, 0, 14569818292224, 0, 1609060073472, 0, 76187049984, 0, 1856540160, 0, 24835584, 0, 182628, 0, 684, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 684*x^16 + 182628*x^14 + 24835584*x^12 + 1856540160*x^10 + 76187049984*x^8 + 1609060073472*x^6 + 14569818292224*x^4 + 27472624091136*x^2 + 163208757248)
 
gp: K = bnfinit(x^18 + 684*x^16 + 182628*x^14 + 24835584*x^12 + 1856540160*x^10 + 76187049984*x^8 + 1609060073472*x^6 + 14569818292224*x^4 + 27472624091136*x^2 + 163208757248, 1)
 

Normalized defining polynomial

\( x^{18} + 684 x^{16} + 182628 x^{14} + 24835584 x^{12} + 1856540160 x^{10} + 76187049984 x^{8} + 1609060073472 x^{6} + 14569818292224 x^{4} + 27472624091136 x^{2} + 163208757248 \)

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:  \(-724363080171793764995208566309796540433577438871552=-\,2^{27}\cdot 3^{44}\cdot 19^{17}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $669.20$
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:  $2, 3, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4104=2^{3}\cdot 3^{3}\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{4104}(1,·)$, $\chi_{4104}(901,·)$, $\chi_{4104}(2113,·)$, $\chi_{4104}(3721,·)$, $\chi_{4104}(3661,·)$, $\chi_{4104}(3757,·)$, $\chi_{4104}(1873,·)$, $\chi_{4104}(3613,·)$, $\chi_{4104}(2605,·)$, $\chi_{4104}(3361,·)$, $\chi_{4104}(1573,·)$, $\chi_{4104}(3313,·)$, $\chi_{4104}(3049,·)$, $\chi_{4104}(3373,·)$, $\chi_{4104}(829,·)$, $\chi_{4104}(1393,·)$, $\chi_{4104}(841,·)$, $\chi_{4104}(1405,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{16} a^{4} - \frac{1}{8} a^{2}$, $\frac{1}{64} a^{5} - \frac{1}{32} a^{3} + \frac{1}{4} a$, $\frac{1}{128} a^{6} - \frac{1}{64} a^{4} + \frac{1}{8} a^{2}$, $\frac{1}{128} a^{7} + \frac{3}{32} a^{3} + \frac{1}{4} a$, $\frac{1}{1024} a^{8} - \frac{1}{256} a^{6} + \frac{5}{256} a^{4} - \frac{1}{32} a^{2}$, $\frac{1}{4096} a^{9} + \frac{1}{1024} a^{7} + \frac{1}{1024} a^{5} + \frac{3}{128} a^{3}$, $\frac{1}{16384} a^{10} + \frac{1}{4096} a^{8} + \frac{1}{4096} a^{6} + \frac{3}{512} a^{4}$, $\frac{1}{65536} a^{11} - \frac{1}{16384} a^{9} - \frac{7}{16384} a^{7} - \frac{7}{1024} a^{5} + \frac{1}{256} a^{3} - \frac{1}{8} a$, $\frac{1}{48234496} a^{12} + \frac{319}{12058624} a^{10} - \frac{4743}{12058624} a^{8} + \frac{889}{753664} a^{6} - \frac{2463}{188416} a^{4} + \frac{407}{5888} a^{2} - \frac{21}{46}$, $\frac{1}{96468992} a^{13} - \frac{49}{24117248} a^{11} + \frac{2617}{24117248} a^{9} + \frac{3005}{1507328} a^{7} + \frac{481}{376832} a^{5} + \frac{637}{11776} a^{3} - \frac{19}{184} a$, $\frac{1}{2378539466752} a^{14} + \frac{4397}{594634866688} a^{12} - \frac{5470271}{594634866688} a^{10} + \frac{10408881}{74329358336} a^{8} + \frac{33921055}{9291169792} a^{6} - \frac{27743693}{1161396224} a^{4} - \frac{6770399}{36293632} a^{2} - \frac{21153}{283544}$, $\frac{1}{19028315734016} a^{15} + \frac{16725}{4757078933504} a^{13} - \frac{26033375}{4757078933504} a^{11} + \frac{35613477}{594634866688} a^{9} - \frac{66268601}{74329358336} a^{7} - \frac{8900345}{9291169792} a^{5} + \frac{26361101}{290349056} a^{3} - \frac{717685}{2268352} a$, $\frac{1}{8877328699652237688832} a^{16} + \frac{160341973}{2219332174913059422208} a^{14} - \frac{14981692277727}{2219332174913059422208} a^{12} - \frac{328531152656667}{277416521864132427776} a^{10} + \frac{8438499079137911}{34677065233016553472} a^{8} - \frac{8624638693685337}{4334633154127069184} a^{6} + \frac{1553124959853337}{135457286066470912} a^{4} + \frac{2635679001297}{33070626481072} a^{2} + \frac{452664995975}{8267656620268}$, $\frac{1}{17754657399304475377664} a^{17} - \frac{72924303}{4438664349826118844416} a^{15} + \frac{15424100266321}{4438664349826118844416} a^{13} + \frac{1580546059317107}{554833043728264855552} a^{11} - \frac{5121648952340003}{69354130466033106944} a^{9} - \frac{25942928734284279}{8669266308254138368} a^{7} - \frac{445275260799405}{541829144265883648} a^{5} - \frac{13024180810855}{252718817288192} a^{3} + \frac{64885266681629}{132282505924288} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{36}\times C_{781596}$, which has order $3601594368$ (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:  \( -1 \) (order $2$)
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:  \( 373976932305.4817 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{18}$ (as 18T1):

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

Intermediate fields

\(\Q(\sqrt{-38}) \), 3.3.29241.1, 6.0.8317790995968.2, 9.9.532962204162830310969.9

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

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 $18$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/23.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ $18$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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$2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
$3$3.9.22.1$x^{9} + 18 x^{8} + 21 x^{6} + 18 x^{5} + 9 x^{3} + 15$$9$$1$$22$$C_9$$[2, 3]$
3.9.22.1$x^{9} + 18 x^{8} + 21 x^{6} + 18 x^{5} + 9 x^{3} + 15$$9$$1$$22$$C_9$$[2, 3]$
19Data not computed