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Properties

Label 18.0.33471151027...6288.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 7^{15}\cdot 11^{6}\cdot 19^{15}$
Root discriminant $261.84$
Ramified primes $2, 7, 11, 19$
Class number $27396096$ (GRH)
Class group $[2, 2, 2, 2, 4, 28, 15288]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8569634816, -14371176448, 6471729152, 5673216000, -2724186624, -759789312, 858619904, 137544224, -83825356, -8055840, 6703724, 487528, 272213, 12684, 7335, -352, 139, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 139*x^16 - 352*x^15 + 7335*x^14 + 12684*x^13 + 272213*x^12 + 487528*x^11 + 6703724*x^10 - 8055840*x^9 - 83825356*x^8 + 137544224*x^7 + 858619904*x^6 - 759789312*x^5 - 2724186624*x^4 + 5673216000*x^3 + 6471729152*x^2 - 14371176448*x + 8569634816)
 
gp: K = bnfinit(x^18 - 4*x^17 + 139*x^16 - 352*x^15 + 7335*x^14 + 12684*x^13 + 272213*x^12 + 487528*x^11 + 6703724*x^10 - 8055840*x^9 - 83825356*x^8 + 137544224*x^7 + 858619904*x^6 - 759789312*x^5 - 2724186624*x^4 + 5673216000*x^3 + 6471729152*x^2 - 14371176448*x + 8569634816, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} + 139 x^{16} - 352 x^{15} + 7335 x^{14} + 12684 x^{13} + 272213 x^{12} + 487528 x^{11} + 6703724 x^{10} - 8055840 x^{9} - 83825356 x^{8} + 137544224 x^{7} + 858619904 x^{6} - 759789312 x^{5} - 2724186624 x^{4} + 5673216000 x^{3} + 6471729152 x^{2} - 14371176448 x + 8569634816 \)

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:  \(-33471151027064787570494624202065583768076288=-\,2^{18}\cdot 7^{15}\cdot 11^{6}\cdot 19^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $261.84$
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:  $2, 7, 11, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{8} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{8} + \frac{1}{8} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{9} + \frac{1}{16} a^{5} - \frac{1}{4} a^{4} + \frac{1}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{64} a^{10} - \frac{1}{32} a^{8} - \frac{1}{16} a^{7} - \frac{7}{64} a^{6} - \frac{1}{16} a^{5} - \frac{7}{16} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{64} a^{11} - \frac{1}{32} a^{9} - \frac{1}{16} a^{8} + \frac{1}{64} a^{7} + \frac{1}{16} a^{6} - \frac{1}{8} a^{5} - \frac{1}{8} a^{4} + \frac{5}{16} a^{3}$, $\frac{1}{64} a^{12} - \frac{3}{64} a^{8} - \frac{1}{16} a^{7} - \frac{3}{32} a^{6} + \frac{1}{16} a^{5} - \frac{3}{16} a^{4} - \frac{1}{8} a^{3} + \frac{3}{8} a^{2}$, $\frac{1}{256} a^{13} - \frac{1}{256} a^{11} + \frac{7}{256} a^{9} - \frac{1}{16} a^{8} + \frac{9}{256} a^{7} + \frac{1}{64} a^{5} - \frac{3}{16} a^{4} - \frac{7}{64} a^{3} - \frac{1}{4} a^{2} + \frac{1}{8} a - \frac{1}{2}$, $\frac{1}{4096} a^{14} + \frac{1}{1024} a^{13} - \frac{21}{4096} a^{12} - \frac{1}{512} a^{11} - \frac{25}{4096} a^{10} - \frac{31}{1024} a^{9} + \frac{117}{4096} a^{8} + \frac{9}{256} a^{7} + \frac{27}{1024} a^{6} - \frac{195}{1024} a^{4} + \frac{23}{64} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{7}{16}$, $\frac{1}{8192} a^{15} - \frac{5}{8192} a^{13} + \frac{3}{2048} a^{12} - \frac{25}{8192} a^{11} - \frac{3}{1024} a^{10} + \frac{69}{8192} a^{9} + \frac{95}{2048} a^{8} - \frac{109}{2048} a^{7} - \frac{35}{512} a^{6} + \frac{221}{2048} a^{5} + \frac{15}{512} a^{4} - \frac{29}{64} a^{3} - \frac{1}{16} a^{2} - \frac{11}{32} a - \frac{1}{8}$, $\frac{1}{262144} a^{16} - \frac{1}{32768} a^{15} - \frac{13}{262144} a^{14} + \frac{37}{65536} a^{13} + \frac{943}{262144} a^{12} - \frac{57}{16384} a^{11} - \frac{51}{262144} a^{10} + \frac{301}{65536} a^{9} + \frac{1809}{65536} a^{8} - \frac{553}{16384} a^{7} - \frac{2651}{65536} a^{6} + \frac{245}{16384} a^{5} - \frac{1469}{8192} a^{4} + \frac{171}{512} a^{3} + \frac{409}{1024} a^{2} - \frac{115}{256} a - \frac{1}{128}$, $\frac{1}{13973924699797943185488172376067721587116942215113317969133854565859328} a^{17} - \frac{3077323720963642928866031184164200526464890704349565443546954061}{1746740587474742898186021547008465198389617776889164746141731820732416} a^{16} + \frac{547375303310428553496593460164722228698555650311607836609792013043}{13973924699797943185488172376067721587116942215113317969133854565859328} a^{15} - \frac{1733452447256738723400343953916546266767508713927839789571028131}{3493481174949485796372043094016930396779235553778329492283463641464832} a^{14} + \frac{17684222050265583966138316004875525775933768011741563186919700442159}{13973924699797943185488172376067721587116942215113317969133854565859328} a^{13} + \frac{4748818189107716746321890857924566162045898970307819816213988777325}{873370293737371449093010773504232599194808888444582373070865910366208} a^{12} - \frac{102597361127242216260107945330954836281113127018116068645971638528563}{13973924699797943185488172376067721587116942215113317969133854565859328} a^{11} + \frac{15341417752456565076991139991548522792194195336929254740953880176629}{3493481174949485796372043094016930396779235553778329492283463641464832} a^{10} + \frac{28492478430740602001936318155953927516277378179577137139979119146545}{3493481174949485796372043094016930396779235553778329492283463641464832} a^{9} - \frac{47486755439235559104492615410184188030462418455692189866427478156641}{873370293737371449093010773504232599194808888444582373070865910366208} a^{8} + \frac{187759423497063564979471522737250934423928982589942624466827768964901}{3493481174949485796372043094016930396779235553778329492283463641464832} a^{7} - \frac{7292224079081379953277350341899126852362697028309348533541291845763}{873370293737371449093010773504232599194808888444582373070865910366208} a^{6} - \frac{30078538715441307885149315877339839907380666002326180754989605831341}{436685146868685724546505386752116299597404444222291186535432955183104} a^{5} + \frac{97257046330821367514734031564812080723568226046867038020234137909}{27292821679292857784156586672007268724837777763893199158464559698944} a^{4} + \frac{6342299190360663518103438463730434183651724315455330432624808964633}{54585643358585715568313173344014537449675555527786398316929119397888} a^{3} - \frac{39145999541382206603988386493157014489147709395853084423897568825}{104171075111804800702887735389340720323808312075928241062841830912} a^{2} + \frac{2725837498472872221835134414409613796885744949727747176954583740175}{6823205419823214446039146668001817181209444440973299789616139924736} a + \frac{2882849368578254931644798557826086250990492925902146267734014773}{213225169369475451438723333375056786912795138780415618425504372648}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{28}\times C_{15288}$, which has order $27396096$ (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:  \( 2494653063.4840164 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-133}) \), 3.3.17689.2, 3.3.15884.1, 6.0.2663410937152.2, Deg 6, 9.9.471484994327458496.3

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

Sibling fields

Galois closure: data not computed
Degree 12 sibling: data not computed
Degree 18 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R R ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
7Data not computed
$11$11.6.0.1$x^{6} + x^{2} - 2 x + 8$$1$$6$$0$$C_6$$[\ ]^{6}$
11.12.6.1$x^{12} + 242 x^{8} + 21296 x^{6} + 14641 x^{4} + 1932612 x^{2} + 113379904$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$19$19.6.5.4$x^{6} + 76$$6$$1$$5$$C_6$$[\ ]_{6}$
19.6.5.4$x^{6} + 76$$6$$1$$5$$C_6$$[\ ]_{6}$
19.6.5.4$x^{6} + 76$$6$$1$$5$$C_6$$[\ ]_{6}$