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Properties

Label 18.0.75915107157...0483.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{9}\cdot 199^{12}$
Root discriminant $59.04$
Ramified primes $3, 199$
Class number $27$ (GRH)
Class group $[3, 9]$ (GRH)
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![675, -8235, 41544, -98271, 126072, -86466, 20371, 3225, 7407, -10575, 7494, -3363, 1048, -399, 225, -108, 39, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 225*x^14 - 399*x^13 + 1048*x^12 - 3363*x^11 + 7494*x^10 - 10575*x^9 + 7407*x^8 + 3225*x^7 + 20371*x^6 - 86466*x^5 + 126072*x^4 - 98271*x^3 + 41544*x^2 - 8235*x + 675)
 
gp: K = bnfinit(x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 225*x^14 - 399*x^13 + 1048*x^12 - 3363*x^11 + 7494*x^10 - 10575*x^9 + 7407*x^8 + 3225*x^7 + 20371*x^6 - 86466*x^5 + 126072*x^4 - 98271*x^3 + 41544*x^2 - 8235*x + 675, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 39 x^{16} - 108 x^{15} + 225 x^{14} - 399 x^{13} + 1048 x^{12} - 3363 x^{11} + 7494 x^{10} - 10575 x^{9} + 7407 x^{8} + 3225 x^{7} + 20371 x^{6} - 86466 x^{5} + 126072 x^{4} - 98271 x^{3} + 41544 x^{2} - 8235 x + 675 \)

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:  \(-75915107157998220182825835900483=-\,3^{9}\cdot 199^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.04$
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:  $3, 199$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{2}$, $\frac{1}{15} a^{9} - \frac{2}{15} a^{8} + \frac{1}{15} a^{7} + \frac{4}{15} a^{5} - \frac{1}{5} a^{4} - \frac{1}{5} a^{3} - \frac{7}{15} a^{2} - \frac{2}{5} a$, $\frac{1}{15} a^{10} + \frac{2}{15} a^{8} + \frac{2}{15} a^{7} - \frac{1}{15} a^{6} + \frac{1}{3} a^{5} + \frac{1}{15} a^{4} + \frac{2}{15} a^{3} + \frac{1}{5} a$, $\frac{1}{15} a^{11} + \frac{1}{15} a^{8} + \frac{2}{15} a^{7} - \frac{2}{15} a^{5} + \frac{1}{5} a^{4} - \frac{4}{15} a^{3} + \frac{2}{15} a^{2} - \frac{1}{5} a$, $\frac{1}{45} a^{12} - \frac{1}{45} a^{10} - \frac{1}{15} a^{8} - \frac{1}{15} a^{7} - \frac{1}{45} a^{6} - \frac{7}{15} a^{5} - \frac{17}{45} a^{4} - \frac{4}{15} a^{3} - \frac{2}{15} a^{2} + \frac{2}{5} a$, $\frac{1}{495} a^{13} - \frac{1}{495} a^{12} - \frac{1}{495} a^{11} - \frac{1}{45} a^{10} - \frac{4}{165} a^{9} + \frac{6}{55} a^{8} - \frac{16}{495} a^{7} - \frac{53}{495} a^{6} + \frac{148}{495} a^{5} - \frac{41}{99} a^{4} + \frac{38}{165} a^{3} - \frac{2}{55} a^{2} + \frac{1}{11} a + \frac{5}{11}$, $\frac{1}{495} a^{14} - \frac{2}{495} a^{12} - \frac{4}{165} a^{11} + \frac{2}{99} a^{10} + \frac{1}{55} a^{9} + \frac{1}{99} a^{8} - \frac{4}{55} a^{7} + \frac{62}{495} a^{6} - \frac{8}{165} a^{5} + \frac{41}{495} a^{4} - \frac{26}{55} a^{3} - \frac{8}{55} a^{2} + \frac{8}{55} a + \frac{5}{11}$, $\frac{1}{495} a^{15} - \frac{1}{165} a^{12} + \frac{8}{495} a^{11} + \frac{1}{55} a^{10} + \frac{14}{495} a^{9} + \frac{13}{165} a^{8} - \frac{23}{165} a^{7} - \frac{1}{55} a^{6} + \frac{238}{495} a^{5} - \frac{79}{165} a^{4} - \frac{58}{165} a^{3} - \frac{32}{165} a^{2} - \frac{9}{55} a - \frac{1}{11}$, $\frac{1}{182137727475} a^{16} - \frac{8}{182137727475} a^{15} + \frac{21831562}{182137727475} a^{14} - \frac{152820794}{182137727475} a^{13} - \frac{1735495727}{182137727475} a^{12} + \frac{51425831}{36427545495} a^{11} + \frac{4412414966}{182137727475} a^{10} - \frac{5800042252}{182137727475} a^{9} + \frac{26314655123}{182137727475} a^{8} - \frac{12816066319}{182137727475} a^{7} + \frac{478991386}{16557975225} a^{6} + \frac{59660238347}{182137727475} a^{5} - \frac{119613964}{20237525275} a^{4} + \frac{1286618814}{4047505055} a^{3} + \frac{1715375612}{5519325075} a^{2} - \frac{29874391}{809501011} a + \frac{385474516}{809501011}$, $\frac{1}{4553443186875} a^{17} + \frac{4}{4553443186875} a^{16} + \frac{21831466}{4553443186875} a^{15} + \frac{396938}{16557975225} a^{14} + \frac{92717393}{182137727475} a^{13} - \frac{46693624924}{4553443186875} a^{12} + \frac{1250290901}{413949380625} a^{11} + \frac{7810785446}{910688637375} a^{10} + \frac{104632060109}{4553443186875} a^{9} + \frac{693728010467}{4553443186875} a^{8} + \frac{619398204853}{4553443186875} a^{7} - \frac{23191035686}{4553443186875} a^{6} - \frac{159366171277}{413949380625} a^{5} + \frac{1877763418423}{4553443186875} a^{4} - \frac{404131435643}{1517814395625} a^{3} - \frac{9899983997}{505938131875} a^{2} + \frac{23021710431}{101187626375} a - \frac{8105548981}{20237525275}$

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

Class group and class number

$C_{3}\times C_{9}$, 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:  \( -\frac{284930594}{137983126875} a^{17} + \frac{2421910049}{137983126875} a^{16} - \frac{29778589612}{413949380625} a^{15} + \frac{346776974}{1839775025} a^{14} - \frac{1245077486}{3311595045} a^{13} + \frac{29892962502}{45994375625} a^{12} - \frac{772462250902}{413949380625} a^{11} + \frac{15195361676}{2508784125} a^{10} - \frac{5211559732538}{413949380625} a^{9} + \frac{736120813084}{45994375625} a^{8} - \frac{308018715286}{37631761875} a^{7} - \frac{447123260322}{45994375625} a^{6} - \frac{1777185034336}{37631761875} a^{5} + \frac{21241509331313}{137983126875} a^{4} - \frac{25916556263174}{137983126875} a^{3} + \frac{5518537087754}{45994375625} a^{2} - \frac{345163340317}{9198875125} a + \frac{8776884742}{1839775025} \) (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:  \( 35164430.4379 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.118803.1 x3, 3.3.39601.1, 6.0.42342458427.1, 6.0.42342458427.2, 6.0.1069227.2 x2, 9.3.1676803696167627.2 x3

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

Sibling fields

Degree 6 sibling: 6.0.1069227.2
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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$199$199.3.2.1$x^{3} - 199$$3$$1$$2$$C_3$$[\ ]_{3}$
199.3.2.1$x^{3} - 199$$3$$1$$2$$C_3$$[\ ]_{3}$
199.3.2.1$x^{3} - 199$$3$$1$$2$$C_3$$[\ ]_{3}$
199.3.2.1$x^{3} - 199$$3$$1$$2$$C_3$$[\ ]_{3}$
199.3.2.1$x^{3} - 199$$3$$1$$2$$C_3$$[\ ]_{3}$
199.3.2.1$x^{3} - 199$$3$$1$$2$$C_3$$[\ ]_{3}$