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Properties

Label 18.0.14367036602...7392.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{31}\cdot 7^{6}\cdot 13^{6}$
Root discriminant $47.36$
Ramified primes $2, 3, 7, 13$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_3\times S_3^2$ (as 18T46)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![226233, -418446, 122472, 302292, -319788, 84807, 70551, -99792, 86670, -63180, 38367, -18900, 8037, -3024, 1026, -279, 63, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 63*x^16 - 279*x^15 + 1026*x^14 - 3024*x^13 + 8037*x^12 - 18900*x^11 + 38367*x^10 - 63180*x^9 + 86670*x^8 - 99792*x^7 + 70551*x^6 + 84807*x^5 - 319788*x^4 + 302292*x^3 + 122472*x^2 - 418446*x + 226233)
 
gp: K = bnfinit(x^18 - 9*x^17 + 63*x^16 - 279*x^15 + 1026*x^14 - 3024*x^13 + 8037*x^12 - 18900*x^11 + 38367*x^10 - 63180*x^9 + 86670*x^8 - 99792*x^7 + 70551*x^6 + 84807*x^5 - 319788*x^4 + 302292*x^3 + 122472*x^2 - 418446*x + 226233, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 63 x^{16} - 279 x^{15} + 1026 x^{14} - 3024 x^{13} + 8037 x^{12} - 18900 x^{11} + 38367 x^{10} - 63180 x^{9} + 86670 x^{8} - 99792 x^{7} + 70551 x^{6} + 84807 x^{5} - 319788 x^{4} + 302292 x^{3} + 122472 x^{2} - 418446 x + 226233 \)

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:  \(-1436703660250682047468690747392=-\,2^{12}\cdot 3^{31}\cdot 7^{6}\cdot 13^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.36$
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:  $2, 3, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3} a^{4}$, $\frac{1}{3} a^{5}$, $\frac{1}{3} a^{6}$, $\frac{1}{3} a^{7}$, $\frac{1}{9} a^{8}$, $\frac{1}{9} a^{9}$, $\frac{1}{9} a^{10}$, $\frac{1}{27} a^{11}$, $\frac{1}{189} a^{12} + \frac{1}{189} a^{11} + \frac{2}{63} a^{9} + \frac{2}{63} a^{8} + \frac{1}{7} a^{6} + \frac{1}{7} a^{5} - \frac{1}{7} a^{3} - \frac{1}{7} a^{2}$, $\frac{1}{189} a^{13} - \frac{1}{189} a^{11} + \frac{2}{63} a^{10} - \frac{2}{63} a^{8} + \frac{1}{7} a^{7} - \frac{1}{7} a^{5} - \frac{1}{7} a^{4} + \frac{1}{7} a^{2}$, $\frac{1}{189} a^{14} - \frac{1}{21} a^{8} - \frac{1}{7} a^{2}$, $\frac{1}{567} a^{15} - \frac{1}{63} a^{9} + \frac{2}{7} a^{3}$, $\frac{1}{128709} a^{16} - \frac{50}{128709} a^{15} - \frac{2}{14301} a^{14} - \frac{5}{4767} a^{13} - \frac{1}{4767} a^{12} - \frac{538}{42903} a^{11} + \frac{5}{2043} a^{10} + \frac{97}{4767} a^{9} - \frac{533}{14301} a^{8} - \frac{338}{4767} a^{7} + \frac{131}{1589} a^{6} - \frac{53}{4767} a^{5} + \frac{131}{1589} a^{4} + \frac{80}{227} a^{3} - \frac{527}{1589} a^{2} - \frac{80}{227} a - \frac{29}{227}$, $\frac{1}{844389427898275478588313} a^{17} - \frac{31476220322987285}{93821047544252830954257} a^{16} + \frac{23338746084647837666}{93821047544252830954257} a^{15} + \frac{566726283135299553662}{281463142632758492862771} a^{14} + \frac{281510812333998830248}{281463142632758492862771} a^{13} + \frac{79536952807356379708}{31273682514750943651419} a^{12} - \frac{168850929235211113820}{13403006792036118707751} a^{11} + \frac{474489946472619752855}{13403006792036118707751} a^{10} + \frac{191671109684043953011}{4467668930678706235917} a^{9} - \frac{3841667314217289881617}{93821047544252830954257} a^{8} + \frac{1227000713326482010822}{31273682514750943651419} a^{7} - \frac{1674592289971008200474}{10424560838250314550473} a^{6} - \frac{668786181291284569064}{31273682514750943651419} a^{5} + \frac{1094954725233472882831}{10424560838250314550473} a^{4} - \frac{8497902010372481732}{17005808871533955221} a^{3} + \frac{138070497543042302858}{1489222976892902078639} a^{2} - \frac{9818458333505519440}{1489222976892902078639} a - \frac{9501782206389098398}{212746139556128868377}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( \frac{359265602}{71518409003727} a^{17} - \frac{8101148798}{71518409003727} a^{16} + \frac{17776632224}{23839469667909} a^{15} - \frac{103153815865}{23839469667909} a^{14} + \frac{355867171985}{23839469667909} a^{13} - \frac{1182326109190}{23839469667909} a^{12} + \frac{138852447529}{1135212841329} a^{11} - \frac{117727525826}{378404280443} a^{10} + \frac{2136814036456}{3405638523987} a^{9} - \frac{2764341297605}{2648829963101} a^{8} + \frac{3441885567094}{2648829963101} a^{7} - \frac{3928609620375}{2648829963101} a^{6} + \frac{3023810284260}{2648829963101} a^{5} + \frac{5147827074690}{2648829963101} a^{4} - \frac{19840986799568}{2648829963101} a^{3} + \frac{281533205091}{54057754349} a^{2} + \frac{2546147577405}{378404280443} a - \frac{403873996910}{54057754349} \) (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:  \( 246972334.17984632 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3^2$ (as 18T46):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 27 conjugacy class representatives for $C_3\times S_3^2$
Character table for $C_3\times S_3^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.3.756.1, 6.0.3326427.1, 6.0.1714608.1

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

Sibling fields

Degree 12 sibling: data not computed
Degree 18 siblings: 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
2Data not computed
3Data not computed
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
$13$13.3.2.1$x^{3} + 26$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.6.4.1$x^{6} + 39 x^{3} + 676$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$