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Properties

Label 18.0.64119276021...9375.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{21}\cdot 5^{9}\cdot 11^{12}$
Root discriminant $39.85$
Ramified primes $3, 5, 11$
Class number $18$ (GRH)
Class group $[3, 6]$ (GRH)
Galois group $C_3^2 : C_2$ (as 18T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![118815, -212265, 378474, -394488, 364521, -237165, 148498, -80625, 55638, -40029, 28578, -17001, 8326, -3297, 1059, -276, 60, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 60*x^16 - 276*x^15 + 1059*x^14 - 3297*x^13 + 8326*x^12 - 17001*x^11 + 28578*x^10 - 40029*x^9 + 55638*x^8 - 80625*x^7 + 148498*x^6 - 237165*x^5 + 364521*x^4 - 394488*x^3 + 378474*x^2 - 212265*x + 118815)
 
gp: K = bnfinit(x^18 - 9*x^17 + 60*x^16 - 276*x^15 + 1059*x^14 - 3297*x^13 + 8326*x^12 - 17001*x^11 + 28578*x^10 - 40029*x^9 + 55638*x^8 - 80625*x^7 + 148498*x^6 - 237165*x^5 + 364521*x^4 - 394488*x^3 + 378474*x^2 - 212265*x + 118815, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 60 x^{16} - 276 x^{15} + 1059 x^{14} - 3297 x^{13} + 8326 x^{12} - 17001 x^{11} + 28578 x^{10} - 40029 x^{9} + 55638 x^{8} - 80625 x^{7} + 148498 x^{6} - 237165 x^{5} + 364521 x^{4} - 394488 x^{3} + 378474 x^{2} - 212265 x + 118815 \)

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:  \(-64119276021132037084693359375=-\,3^{21}\cdot 5^{9}\cdot 11^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.85$
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:  $3, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{10} a^{10} - \frac{1}{5} a^{6} - \frac{1}{2} a^{3} - \frac{2}{5} a^{2}$, $\frac{1}{10} a^{11} - \frac{1}{5} a^{7} - \frac{1}{2} a^{4} - \frac{2}{5} a^{3}$, $\frac{1}{20} a^{12} - \frac{1}{20} a^{10} + \frac{3}{20} a^{8} - \frac{3}{20} a^{6} + \frac{1}{20} a^{4} - \frac{1}{2} a^{3} + \frac{9}{20} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{100} a^{13} + \frac{1}{100} a^{12} + \frac{1}{100} a^{11} + \frac{1}{100} a^{10} + \frac{13}{100} a^{9} - \frac{7}{100} a^{8} + \frac{13}{100} a^{7} + \frac{23}{100} a^{6} - \frac{49}{100} a^{5} + \frac{31}{100} a^{4} - \frac{29}{100} a^{3} + \frac{11}{100} a^{2} - \frac{1}{20} a + \frac{7}{20}$, $\frac{1}{80100} a^{14} - \frac{7}{80100} a^{13} - \frac{377}{80100} a^{12} + \frac{2353}{80100} a^{11} + \frac{541}{16020} a^{10} + \frac{4789}{80100} a^{9} + \frac{12259}{80100} a^{8} - \frac{11551}{80100} a^{7} - \frac{12083}{80100} a^{6} + \frac{841}{26700} a^{5} + \frac{8251}{26700} a^{4} - \frac{1883}{8900} a^{3} - \frac{5981}{26700} a^{2} - \frac{407}{1068} a - \frac{7}{15}$, $\frac{1}{240300} a^{15} + \frac{5}{3204} a^{13} + \frac{103}{48060} a^{12} + \frac{1319}{80100} a^{11} - \frac{167}{5340} a^{10} - \frac{4781}{48060} a^{9} - \frac{763}{16020} a^{8} - \frac{3479}{80100} a^{7} - \frac{89}{540} a^{6} + \frac{2881}{16020} a^{5} + \frac{4067}{16020} a^{4} + \frac{8143}{80100} a^{3} + \frac{6733}{16020} a^{2} + \frac{1171}{2670} a - \frac{11}{36}$, $\frac{1}{3206927254500} a^{16} - \frac{2}{801731813625} a^{15} - \frac{851857}{534487875750} a^{14} + \frac{17889067}{1603463627250} a^{13} + \frac{38489273503}{1603463627250} a^{12} - \frac{11803639391}{267243937875} a^{11} + \frac{727369319}{320692725450} a^{10} - \frac{303911492459}{1603463627250} a^{9} - \frac{121414886}{10689757515} a^{8} - \frac{83112056138}{801731813625} a^{7} + \frac{213906938273}{1603463627250} a^{6} - \frac{70747803317}{178162625250} a^{5} + \frac{10652325779}{267243937875} a^{4} + \frac{23232585719}{178162625250} a^{3} + \frac{342494942207}{1068975751500} a^{2} - \frac{21620998037}{53448787575} a - \frac{327468887}{1201096350}$, $\frac{1}{7751143174126500} a^{17} + \frac{4}{25837143913755} a^{16} - \frac{349543793}{3875571587063250} a^{15} - \frac{233432638}{1937785793531625} a^{14} - \frac{682366234951}{287079376819500} a^{13} + \frac{9603397504163}{3875571587063250} a^{12} - \frac{73374961945861}{7751143174126500} a^{11} - \frac{17410029576763}{1291857195687750} a^{10} - \frac{22427758542059}{7751143174126500} a^{9} - \frac{129296524700723}{1937785793531625} a^{8} + \frac{543272280583}{3827725024260} a^{7} - \frac{198834463862632}{1937785793531625} a^{6} - \frac{226964898254417}{516742878275100} a^{5} + \frac{71037009419083}{645928597843875} a^{4} - \frac{120189226253089}{645928597843875} a^{3} + \frac{138007843270201}{430619065229250} a^{2} - \frac{56193004897}{11483175072780} a + \frac{529966416232}{1451524938975}$

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

Class group and class number

$C_{3}\times C_{6}$, which has order $18$ (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:  \( 13204003.8996 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_3$ (as 18T4):

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 $C_3^2 : C_2$
Character table for $C_3^2 : C_2$

Intermediate fields

\(\Q(\sqrt{-15}) \), 3.1.1815.1 x3, 3.1.16335.2 x3, 3.1.135.1 x3, 3.1.16335.1 x3, 6.0.49413375.1, 6.0.4002483375.3, 6.0.273375.1, 6.0.4002483375.2, 9.1.65380565930625.1 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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ R R ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$3.6.7.1$x^{6} + 6 x^{2} + 6$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.1$x^{6} + 6 x^{2} + 6$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.1$x^{6} + 6 x^{2} + 6$$6$$1$$7$$S_3$$[3/2]_{2}$
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
$11$11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$