↑

Properties

Label 18.0.28616291218...0000.3
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{9}\cdot 5^{9}\cdot 127^{14}$
Root discriminant $335.25$
Ramified primes $2, 3, 5, 127$
Class number $895476400$ (GRH)
Class group $[2, 2, 10, 22386910]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2600310784, -3598002432, 2792992512, -1193707296, 298949568, -45713808, 27806472, -19725330, 6180610, -67901, -448329, 55770, 35022, -10860, -48, 408, -32, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 - 32*x^16 + 408*x^15 - 48*x^14 - 10860*x^13 + 35022*x^12 + 55770*x^11 - 448329*x^10 - 67901*x^9 + 6180610*x^8 - 19725330*x^7 + 27806472*x^6 - 45713808*x^5 + 298949568*x^4 - 1193707296*x^3 + 2792992512*x^2 - 3598002432*x + 2600310784)
 
gp: K = bnfinit(x^18 - 7*x^17 - 32*x^16 + 408*x^15 - 48*x^14 - 10860*x^13 + 35022*x^12 + 55770*x^11 - 448329*x^10 - 67901*x^9 + 6180610*x^8 - 19725330*x^7 + 27806472*x^6 - 45713808*x^5 + 298949568*x^4 - 1193707296*x^3 + 2792992512*x^2 - 3598002432*x + 2600310784, 1)
 

Normalized defining polynomial

\( x^{18} - 7 x^{17} - 32 x^{16} + 408 x^{15} - 48 x^{14} - 10860 x^{13} + 35022 x^{12} + 55770 x^{11} - 448329 x^{10} - 67901 x^{9} + 6180610 x^{8} - 19725330 x^{7} + 27806472 x^{6} - 45713808 x^{5} + 298949568 x^{4} - 1193707296 x^{3} + 2792992512 x^{2} - 3598002432 x + 2600310784 \)

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:  \(-2861629121862889270656120891466827264000000000=-\,2^{18}\cdot 3^{9}\cdot 5^{9}\cdot 127^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $335.25$
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:  $2, 3, 5, 127$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{8} a^{7} - \frac{1}{16} a^{6} + \frac{1}{16} a^{5} - \frac{1}{4} a^{4} + \frac{1}{8} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{9} + \frac{1}{16} a^{7} + \frac{1}{16} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{64} a^{12} - \frac{1}{32} a^{11} - \frac{1}{64} a^{10} + \frac{1}{64} a^{8} + \frac{3}{32} a^{7} + \frac{5}{64} a^{6} + \frac{3}{16} a^{5} - \frac{7}{32} a^{4} - \frac{1}{4} a^{3} + \frac{1}{8} a^{2}$, $\frac{1}{64} a^{13} - \frac{1}{64} a^{11} - \frac{1}{32} a^{10} - \frac{3}{64} a^{9} + \frac{5}{64} a^{7} + \frac{3}{32} a^{6} + \frac{7}{32} a^{5} - \frac{1}{16} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{128} a^{14} - \frac{1}{128} a^{13} - \frac{1}{128} a^{12} + \frac{3}{128} a^{11} - \frac{1}{128} a^{10} - \frac{1}{128} a^{9} - \frac{3}{128} a^{8} + \frac{5}{128} a^{7} + \frac{1}{16} a^{6} - \frac{15}{64} a^{5} + \frac{3}{32} a^{4} + \frac{3}{16} a^{3} - \frac{1}{8} a^{2}$, $\frac{1}{128} a^{15} - \frac{1}{32} a^{11} - \frac{1}{32} a^{10} - \frac{1}{64} a^{9} + \frac{3}{128} a^{7} + \frac{3}{32} a^{6} + \frac{5}{64} a^{5} + \frac{3}{16} a^{4} - \frac{1}{16} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{117248} a^{16} - \frac{3}{916} a^{15} - \frac{33}{58624} a^{14} + \frac{289}{58624} a^{13} - \frac{279}{58624} a^{12} - \frac{409}{58624} a^{11} - \frac{5}{29312} a^{10} - \frac{2199}{58624} a^{9} + \frac{2077}{117248} a^{8} + \frac{4733}{58624} a^{7} - \frac{4737}{58624} a^{6} - \frac{2219}{29312} a^{5} - \frac{1043}{14656} a^{4} + \frac{2365}{7328} a^{3} - \frac{28}{229} a^{2} + \frac{39}{916} a + \frac{111}{229}$, $\frac{1}{3166716171867992462462227223900066365487643447808} a^{17} - \frac{1097835104517401007212462756334652991894735}{3166716171867992462462227223900066365487643447808} a^{16} - \frac{5678127202118143576614615130421771088326966107}{1583358085933996231231113611950033182743821723904} a^{15} - \frac{132017182567866722209801897123529057955134745}{791679042966998115615556805975016591371910861952} a^{14} + \frac{1253297960898426073900368644199290684986000665}{395839521483499057807778402987508295685955430976} a^{13} + \frac{4639352270216812604061810950353392103832805659}{791679042966998115615556805975016591371910861952} a^{12} - \frac{19265702041946538069077897386698443682689841089}{1583358085933996231231113611950033182743821723904} a^{11} + \frac{13313520176077203721663875175092044386490239413}{1583358085933996231231113611950033182743821723904} a^{10} - \frac{197762920782584969124557257347536202853642523605}{3166716171867992462462227223900066365487643447808} a^{9} - \frac{192917526881833782358174433611975808440324440645}{3166716171867992462462227223900066365487643447808} a^{8} + \frac{41265397991878554828804999185838336973489490997}{395839521483499057807778402987508295685955430976} a^{7} + \frac{174292121491030956094589978944788099847709230805}{1583358085933996231231113611950033182743821723904} a^{6} + \frac{187226569565702393547781098440653144329799492179}{791679042966998115615556805975016591371910861952} a^{5} - \frac{63202975902758080935535812436441472294775496369}{395839521483499057807778402987508295685955430976} a^{4} + \frac{18392290709043567146692177262047697869739069549}{197919760741749528903889201493754147842977715488} a^{3} + \frac{4997161616748712878596038205121791403911864291}{24739970092718691112986150186719268480372214436} a^{2} + \frac{1714594392590448030260880287916198849201367271}{24739970092718691112986150186719268480372214436} a - \frac{1431887840044262309351084217245800457346678507}{6184992523179672778246537546679817120093053609}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{10}\times C_{22386910}$, which has order $895476400$ (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:  \( 5546046730.2947445 \) (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

\(\Q(\sqrt{-15}) \), 3.3.16129.1, 3.3.1016.1, 6.0.3483864000.3, 6.0.877988163375.2, 9.9.272832440404737536.1

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 R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$$0$Trivial$[\ ]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
$3$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$5$5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$127$127.6.4.1$x^{6} + 1016 x^{3} + 435483$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
127.12.10.1$x^{12} - 1270 x^{6} + 11758041$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$