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Properties

Label 18.0.17402447690...6832.3
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{30}\cdot 7^{9}\cdot 19^{14}$
Root discriminant $326.11$
Ramified primes $2, 3, 7, 19$
Class number $392145180$ (GRH)
Class group $[2, 196072590]$ (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![415009792, -911283456, 985387392, -606480096, 236050848, -92066016, 69765816, -46463802, 15726006, -248865, -1661769, 431202, 37830, -32412, 3168, 600, -108, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 108*x^16 + 600*x^15 + 3168*x^14 - 32412*x^13 + 37830*x^12 + 431202*x^11 - 1661769*x^10 - 248865*x^9 + 15726006*x^8 - 46463802*x^7 + 69765816*x^6 - 92066016*x^5 + 236050848*x^4 - 606480096*x^3 + 985387392*x^2 - 911283456*x + 415009792)
 
gp: K = bnfinit(x^18 - 3*x^17 - 108*x^16 + 600*x^15 + 3168*x^14 - 32412*x^13 + 37830*x^12 + 431202*x^11 - 1661769*x^10 - 248865*x^9 + 15726006*x^8 - 46463802*x^7 + 69765816*x^6 - 92066016*x^5 + 236050848*x^4 - 606480096*x^3 + 985387392*x^2 - 911283456*x + 415009792, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 108 x^{16} + 600 x^{15} + 3168 x^{14} - 32412 x^{13} + 37830 x^{12} + 431202 x^{11} - 1661769 x^{10} - 248865 x^{9} + 15726006 x^{8} - 46463802 x^{7} + 69765816 x^{6} - 92066016 x^{5} + 236050848 x^{4} - 606480096 x^{3} + 985387392 x^{2} - 911283456 x + 415009792 \)

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:  \(-1740244769080179953822298827108004668139896832=-\,2^{18}\cdot 3^{30}\cdot 7^{9}\cdot 19^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $326.11$
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:  $2, 3, 7, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{48} a^{10} - \frac{1}{16} a^{9} + \frac{1}{24} a^{8} - \frac{1}{8} a^{7} - \frac{5}{48} a^{6} + \frac{1}{16} a^{5} + \frac{5}{24} a^{4} + \frac{1}{8} a^{3} + \frac{1}{6} a^{2} + \frac{1}{3}$, $\frac{1}{48} a^{11} - \frac{1}{48} a^{9} + \frac{1}{48} a^{7} + \frac{1}{48} a^{5} + \frac{7}{24} a^{3} + \frac{1}{3} a$, $\frac{1}{192} a^{12} - \frac{1}{96} a^{11} - \frac{1}{192} a^{10} + \frac{1}{24} a^{9} + \frac{1}{192} a^{8} + \frac{11}{96} a^{7} + \frac{13}{192} a^{6} - \frac{5}{48} a^{5} + \frac{13}{96} a^{4} - \frac{5}{24} a^{3} + \frac{11}{24} a^{2} - \frac{1}{6} a$, $\frac{1}{192} a^{13} - \frac{1}{192} a^{11} - \frac{1}{96} a^{10} - \frac{11}{192} a^{9} + \frac{1}{24} a^{8} + \frac{13}{192} a^{7} - \frac{1}{96} a^{6} - \frac{17}{96} a^{5} - \frac{5}{48} a^{4} - \frac{1}{6} a^{3} - \frac{1}{12} a^{2} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{25728} a^{14} - \frac{65}{25728} a^{13} + \frac{17}{8576} a^{12} + \frac{187}{25728} a^{11} + \frac{11}{25728} a^{10} + \frac{623}{25728} a^{9} - \frac{381}{8576} a^{8} + \frac{751}{8576} a^{7} - \frac{517}{6432} a^{6} + \frac{1481}{12864} a^{5} - \frac{73}{6432} a^{4} + \frac{439}{3216} a^{3} + \frac{443}{1608} a^{2} - \frac{121}{402} a + \frac{1}{3}$, $\frac{1}{25728} a^{15} - \frac{5}{6432} a^{13} + \frac{3}{4288} a^{12} - \frac{7}{6432} a^{11} + \frac{133}{12864} a^{10} - \frac{253}{4288} a^{9} + \frac{293}{12864} a^{8} + \frac{595}{25728} a^{7} + \frac{1003}{12864} a^{6} + \frac{2185}{12864} a^{5} - \frac{115}{6432} a^{4} - \frac{1265}{3216} a^{3} - \frac{767}{1608} a^{2} - \frac{31}{134} a + \frac{1}{3}$, $\frac{1}{102912} a^{16} - \frac{1}{51456} a^{14} - \frac{107}{51456} a^{13} - \frac{91}{51456} a^{12} - \frac{87}{17152} a^{11} - \frac{43}{8576} a^{10} + \frac{1051}{17152} a^{9} - \frac{3899}{102912} a^{8} + \frac{1649}{51456} a^{7} - \frac{1947}{17152} a^{6} - \frac{263}{8576} a^{5} + \frac{2299}{12864} a^{4} - \frac{703}{2144} a^{3} - \frac{3}{536} a^{2} - \frac{17}{804} a + \frac{1}{3}$, $\frac{1}{1728569323695802220487680786796801142293504} a^{17} + \frac{2733780246918433846751742115654599569}{1728569323695802220487680786796801142293504} a^{16} + \frac{2966049532658225178037498578492180071}{288094887282633703414613464466133523715584} a^{15} + \frac{408557176412328728344963286539609757}{72023721820658425853653366116533380928896} a^{14} + \frac{328471832380742673733361601372706946349}{144047443641316851707306732233066761857792} a^{13} - \frac{146616351902953908697283022799901247379}{72023721820658425853653366116533380928896} a^{12} - \frac{4401211998516499966906927884453920414027}{864284661847901110243840393398400571146752} a^{11} + \frac{352690553657780537258847527099707592417}{288094887282633703414613464466133523715584} a^{10} + \frac{13531563418110735559276284701941758251965}{576189774565267406829226928932267047431168} a^{9} - \frac{49552134157031053166932414838088787249265}{1728569323695802220487680786796801142293504} a^{8} - \frac{6295712733391463696913087669410755641203}{144047443641316851707306732233066761857792} a^{7} + \frac{34439431826393786691206808868839324953401}{864284661847901110243840393398400571146752} a^{6} - \frac{71349044917771280777906007873694918830965}{432142330923950555121920196699200285573376} a^{5} - \frac{42906822411041893677116481070262268099385}{216071165461975277560960098349600142786688} a^{4} + \frac{14073679332156740321145059212652709109907}{36011860910329212926826683058266690464448} a^{3} + \frac{1052000736800730149000818702755857098895}{3376111960343363711890001536712502231042} a^{2} + \frac{407992697979826237451863250368777586003}{4501482613791151615853335382283336308056} a + \frac{1193668138425241441465466679149951851}{50389730751393488237164202040485107926}$

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

Class group and class number

$C_{2}\times C_{196072590}$, which has order $392145180$ (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:  \( 15773424688.63964 \) (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

\(\Q(\sqrt{-7}) \), 3.3.4104.1, 3.3.29241.1, 6.0.293277375783.2, 6.0.5777085888.7, 9.9.6566954215853707776.1

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 R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
3Data not computed
$7$7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$19$19.6.4.3$x^{6} + 95 x^{3} + 2888$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
19.12.10.1$x^{12} - 171 x^{6} + 23104$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$