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Properties

Label 18.0.15816733132...2848.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{33}\cdot 11^{12}\cdot 19^{12}$
Root discriminant $418.96$
Ramified primes $2, 3, 11, 19$
Class number $6561$ (GRH)
Class group $[3, 3, 3, 3, 9, 9]$ (GRH)
Galois group $C_3\times C_3:S_3$ (as 18T23)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2314764156, 374601564, -952674696, 514036110, 146010546, -130560552, 48365515, 1561365, -2055975, 663882, 89241, -44235, 3940, 4365, -1191, 90, 39, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 39*x^16 + 90*x^15 - 1191*x^14 + 4365*x^13 + 3940*x^12 - 44235*x^11 + 89241*x^10 + 663882*x^9 - 2055975*x^8 + 1561365*x^7 + 48365515*x^6 - 130560552*x^5 + 146010546*x^4 + 514036110*x^3 - 952674696*x^2 + 374601564*x + 2314764156)
 
gp: K = bnfinit(x^18 - 9*x^17 + 39*x^16 + 90*x^15 - 1191*x^14 + 4365*x^13 + 3940*x^12 - 44235*x^11 + 89241*x^10 + 663882*x^9 - 2055975*x^8 + 1561365*x^7 + 48365515*x^6 - 130560552*x^5 + 146010546*x^4 + 514036110*x^3 - 952674696*x^2 + 374601564*x + 2314764156, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 39 x^{16} + 90 x^{15} - 1191 x^{14} + 4365 x^{13} + 3940 x^{12} - 44235 x^{11} + 89241 x^{10} + 663882 x^{9} - 2055975 x^{8} + 1561365 x^{7} + 48365515 x^{6} - 130560552 x^{5} + 146010546 x^{4} + 514036110 x^{3} - 952674696 x^{2} + 374601564 x + 2314764156 \)

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:  \(-158167331320896150779524753864609204299658702848=-\,2^{12}\cdot 3^{33}\cdot 11^{12}\cdot 19^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $418.96$
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:  $2, 3, 11, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5}$, $\frac{1}{1646426} a^{15} + \frac{319255}{1646426} a^{14} - \frac{68461}{1646426} a^{13} + \frac{147050}{823213} a^{12} - \frac{325759}{1646426} a^{11} - \frac{613657}{1646426} a^{10} + \frac{319387}{823213} a^{9} + \frac{71791}{1646426} a^{8} + \frac{14259}{44498} a^{7} + \frac{361345}{823213} a^{6} - \frac{442331}{1646426} a^{5} - \frac{732389}{1646426} a^{4} - \frac{544075}{1646426} a^{3} - \frac{216512}{823213} a^{2} + \frac{76116}{823213} a + \frac{331575}{823213}$, $\frac{1}{71492200674917916713938} a^{16} + \frac{3268131009082262}{35746100337458958356969} a^{15} + \frac{173697135093209774744}{1881373701971524124051} a^{14} - \frac{6922616131010564201540}{35746100337458958356969} a^{13} - \frac{2424727641302402229014}{35746100337458958356969} a^{12} - \frac{17850659352205323606144}{35746100337458958356969} a^{11} - \frac{14532954903527514068806}{35746100337458958356969} a^{10} - \frac{16870997139269547528647}{35746100337458958356969} a^{9} + \frac{1808921469963430522553}{35746100337458958356969} a^{8} + \frac{5279142221335678231813}{35746100337458958356969} a^{7} + \frac{7491085623567222740556}{35746100337458958356969} a^{6} - \frac{17013428793693773271899}{35746100337458958356969} a^{5} - \frac{65179676726855611675}{1932221639862646397674} a^{4} + \frac{14813886789935491803220}{35746100337458958356969} a^{3} - \frac{14576733426197190407907}{35746100337458958356969} a^{2} - \frac{17866219050993281918261}{35746100337458958356969} a + \frac{8177837276186494869681}{35746100337458958356969}$, $\frac{1}{39309261078656626736983507517731516880718694} a^{17} + \frac{23128586077607650144}{19654630539328313368491753758865758440359347} a^{16} + \frac{3877311970017870774556074679192731947}{19654630539328313368491753758865758440359347} a^{15} - \frac{7879281760875121871636345412159389948362019}{39309261078656626736983507517731516880718694} a^{14} - \frac{4364169556239052615068826784506165544336875}{39309261078656626736983507517731516880718694} a^{13} + \frac{2379542737097569739748210668420973442185286}{19654630539328313368491753758865758440359347} a^{12} + \frac{13936514853891523418351491990423201642718869}{39309261078656626736983507517731516880718694} a^{11} + \frac{12894610524145626465378941766704090106529171}{39309261078656626736983507517731516880718694} a^{10} + \frac{2380248530623985145120966115985636378244527}{19654630539328313368491753758865758440359347} a^{9} + \frac{139678402018727809023694368079643889849931}{1062412461585314236134689392371122077857262} a^{8} + \frac{17588246487424554472442543918098804594091841}{39309261078656626736983507517731516880718694} a^{7} + \frac{4119996758478635201282389267988325197538063}{19654630539328313368491753758865758440359347} a^{6} + \frac{920163791170837114146455436232847217656794}{19654630539328313368491753758865758440359347} a^{5} + \frac{17696314348260915788213806032830120836221337}{39309261078656626736983507517731516880718694} a^{4} + \frac{289925454512884992997290391873059803269135}{1034454238912016493078513355729776760018913} a^{3} + \frac{338396537200723309451797598266546168002199}{19654630539328313368491753758865758440359347} a^{2} + \frac{2760640350645624148027262515317496259733632}{19654630539328313368491753758865758440359347} a - \frac{1193923419051501737257689556113090172746502}{19654630539328313368491753758865758440359347}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{9}\times C_{9}$, which has order $6561$ (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:  \( \frac{527420400312068467509}{353889297237261130068694918229} a^{17} - \frac{12424317215209625221269}{707778594474522260137389836458} a^{16} + \frac{33215785071552275344011}{353889297237261130068694918229} a^{15} - \frac{18043937962238384343735}{707778594474522260137389836458} a^{14} - \frac{1469624195816635460886579}{707778594474522260137389836458} a^{13} + \frac{7544091494556947726438931}{707778594474522260137389836458} a^{12} - \frac{7579014907040544122637783}{707778594474522260137389836458} a^{11} - \frac{51887887739548954616849007}{707778594474522260137389836458} a^{10} + \frac{173165367606567849796635041}{707778594474522260137389836458} a^{9} + \frac{486360020253097650058904535}{707778594474522260137389836458} a^{8} - \frac{3778987672854892817334202053}{707778594474522260137389836458} a^{7} + \frac{6195840526144897211949169815}{707778594474522260137389836458} a^{6} + \frac{42963118329204242925834970905}{707778594474522260137389836458} a^{5} - \frac{127402795225830201699866759784}{353889297237261130068694918229} a^{4} + \frac{467440005678363402177401735269}{707778594474522260137389836458} a^{3} - \frac{2957553931869083083716203229}{18625752486171638424668153591} a^{2} - \frac{593708211083316316087655706276}{353889297237261130068694918229} a + \frac{846949210343195359364723891356}{353889297237261130068694918229} \) (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:  \( 31268530013404.87 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_3:S_3$ (as 18T23):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 18 conjugacy class representatives for $C_3\times C_3:S_3$
Character table for $C_3\times C_3:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.1083.1 x3, 6.0.41497747632.7, Deg 6, Deg 6, 6.0.3518667.2

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

Sibling fields

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
$3$3.6.11.14$x^{6} + 12 x^{3} + 18 x^{2} + 3$$6$$1$$11$$S_3\times C_3$$[2, 5/2]_{2}$
3.6.11.14$x^{6} + 12 x^{3} + 18 x^{2} + 3$$6$$1$$11$$S_3\times C_3$$[2, 5/2]_{2}$
3.6.11.14$x^{6} + 12 x^{3} + 18 x^{2} + 3$$6$$1$$11$$S_3\times C_3$$[2, 5/2]_{2}$
$11$11.6.4.2$x^{6} - 11 x^{3} + 847$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
11.6.4.2$x^{6} - 11 x^{3} + 847$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
11.6.4.2$x^{6} - 11 x^{3} + 847$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
$19$19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$