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Properties

Label 18.0.56320749286...0448.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{9}\cdot 127^{10}$
Root discriminant $51.09$
Ramified primes $2, 3, 127$
Class number $2$ (GRH)
Class group $[2]$ (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![4096, 26624, 60672, 60160, 30080, 7808, 7792, 3344, 1576, 3160, 1072, 24, 326, -182, -63, 27, -2, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 2*x^16 + 27*x^15 - 63*x^14 - 182*x^13 + 326*x^12 + 24*x^11 + 1072*x^10 + 3160*x^9 + 1576*x^8 + 3344*x^7 + 7792*x^6 + 7808*x^5 + 30080*x^4 + 60160*x^3 + 60672*x^2 + 26624*x + 4096)
 
gp: K = bnfinit(x^18 - 3*x^17 - 2*x^16 + 27*x^15 - 63*x^14 - 182*x^13 + 326*x^12 + 24*x^11 + 1072*x^10 + 3160*x^9 + 1576*x^8 + 3344*x^7 + 7792*x^6 + 7808*x^5 + 30080*x^4 + 60160*x^3 + 60672*x^2 + 26624*x + 4096, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 2 x^{16} + 27 x^{15} - 63 x^{14} - 182 x^{13} + 326 x^{12} + 24 x^{11} + 1072 x^{10} + 3160 x^{9} + 1576 x^{8} + 3344 x^{7} + 7792 x^{6} + 7808 x^{5} + 30080 x^{4} + 60160 x^{3} + 60672 x^{2} + 26624 x + 4096 \)

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:  \(-5632074928630950758566400360448=-\,2^{18}\cdot 3^{9}\cdot 127^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.09$
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:  $2, 3, 127$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{10} - \frac{1}{8} a^{9} + \frac{1}{16} a^{8} - \frac{1}{16} a^{7} + \frac{1}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a$, $\frac{1}{32} a^{12} - \frac{1}{32} a^{11} - \frac{1}{16} a^{10} + \frac{1}{32} a^{9} + \frac{3}{32} a^{8} + \frac{1}{16} a^{7} - \frac{1}{8} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{13} - \frac{1}{32} a^{11} + \frac{1}{32} a^{10} - \frac{1}{8} a^{9} - \frac{1}{32} a^{8} + \frac{1}{8} a^{6} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{64} a^{14} - \frac{1}{64} a^{13} - \frac{1}{64} a^{11} - \frac{1}{64} a^{10} - \frac{1}{16} a^{9} + \frac{1}{32} a^{8} - \frac{1}{16} a^{7} + \frac{1}{8} a^{6} - \frac{1}{8} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{256} a^{15} + \frac{1}{256} a^{14} - \frac{1}{128} a^{13} - \frac{1}{256} a^{12} + \frac{5}{256} a^{11} - \frac{7}{128} a^{10} - \frac{11}{128} a^{9} - \frac{1}{32} a^{8} - \frac{1}{32} a^{7} - \frac{5}{32} a^{6} - \frac{3}{32} a^{5} - \frac{3}{16} a^{4} - \frac{7}{16} a^{3} - \frac{1}{4} a$, $\frac{1}{512} a^{16} - \frac{1}{512} a^{15} - \frac{1}{128} a^{14} + \frac{3}{512} a^{13} + \frac{7}{512} a^{12} + \frac{1}{64} a^{11} - \frac{13}{256} a^{10} + \frac{9}{128} a^{9} + \frac{5}{64} a^{8} - \frac{7}{64} a^{7} - \frac{9}{64} a^{6} - \frac{1}{32} a^{4} + \frac{3}{16} a^{3} - \frac{1}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{1114142393256430556007698432} a^{17} + \frac{982471250617832094154817}{1114142393256430556007698432} a^{16} - \frac{537396447595166435428639}{557071196628215278003849216} a^{15} - \frac{5611907270319372120000445}{1114142393256430556007698432} a^{14} + \frac{5541729427679842527462829}{1114142393256430556007698432} a^{13} + \frac{3422066601920083140438399}{557071196628215278003849216} a^{12} + \frac{4019480469618146891729999}{557071196628215278003849216} a^{11} + \frac{2577593560332184340853643}{69633899578526909750481152} a^{10} + \frac{6023160507833522407020555}{69633899578526909750481152} a^{9} + \frac{5379577101790059203201787}{139267799157053819500962304} a^{8} + \frac{10216248075504577374019089}{139267799157053819500962304} a^{7} - \frac{11387633781411659091615837}{69633899578526909750481152} a^{6} - \frac{95485542878814627062051}{548298421878164643704576} a^{5} + \frac{3487382479987989592040961}{17408474894631727437620288} a^{4} - \frac{1989275850404078112608039}{8704237447315863718810144} a^{3} - \frac{1409776984945322315533285}{4352118723657931859405072} a^{2} + \frac{1812506547499303022917785}{4352118723657931859405072} a - \frac{229499798268852977675291}{1088029680914482964851268}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( -\frac{693570908493955}{432739011821809664} a^{17} + \frac{2340308272355869}{432739011821809664} a^{16} + \frac{262576486421757}{216369505910904832} a^{15} - \frac{18976322657208169}{432739011821809664} a^{14} + \frac{50802472029220505}{432739011821809664} a^{13} + \frac{53823693559736835}{216369505910904832} a^{12} - \frac{133874644475789629}{216369505910904832} a^{11} + \frac{5121641378728171}{27046188238863104} a^{10} - \frac{47915558313508693}{27046188238863104} a^{9} - \frac{238981647733541057}{54092376477726208} a^{8} - \frac{45313337058619219}{54092376477726208} a^{7} - \frac{133093303032384041}{27046188238863104} a^{6} - \frac{290897290481288057}{27046188238863104} a^{5} - \frac{56579125697596667}{6761547059715776} a^{4} - \frac{152010996061114863}{3380773529857888} a^{3} - \frac{134790296096409437}{1690386764928944} a^{2} - \frac{112464181087348891}{1690386764928944} a - \frac{6773562299056895}{422596691232236} \) (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:  \( 621058194.3189892 \) (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

\(\Q(\sqrt{-3}) \), 3.3.1016.1, 6.0.435483.1, 6.0.27870912.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
$3$3.6.3.2$x^{6} - 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}$
$127$$\Q_{127}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{127}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{127}$$x + 9$$1$$1$$0$Trivial$[\ ]$
127.2.1.1$x^{2} - 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.1$x^{2} - 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.1$x^{2} - 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.3.2.1$x^{3} - 127$$3$$1$$2$$C_3$$[\ ]_{3}$
127.6.5.1$x^{6} - 127$$6$$1$$5$$C_6$$[\ ]_{6}$