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Properties

Label 18.18.1503855547...7456.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{20}\cdot 3^{14}\cdot 7^{10}\cdot 101^{6}$
Root discriminant $69.69$
Ramified primes $2, 3, 7, 101$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3:S_3:S_4$ (as 18T154)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3136, 9184, 140512, -718892, 830632, 612252, -1264603, 53790, 611032, -165800, -126883, 54168, 10378, -7212, 5, 406, -34, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 8*x^17 - 34*x^16 + 406*x^15 + 5*x^14 - 7212*x^13 + 10378*x^12 + 54168*x^11 - 126883*x^10 - 165800*x^9 + 611032*x^8 + 53790*x^7 - 1264603*x^6 + 612252*x^5 + 830632*x^4 - 718892*x^3 + 140512*x^2 + 9184*x - 3136)
 
gp: K = bnfinit(x^18 - 8*x^17 - 34*x^16 + 406*x^15 + 5*x^14 - 7212*x^13 + 10378*x^12 + 54168*x^11 - 126883*x^10 - 165800*x^9 + 611032*x^8 + 53790*x^7 - 1264603*x^6 + 612252*x^5 + 830632*x^4 - 718892*x^3 + 140512*x^2 + 9184*x - 3136, 1)
 

Normalized defining polynomial

\( x^{18} - 8 x^{17} - 34 x^{16} + 406 x^{15} + 5 x^{14} - 7212 x^{13} + 10378 x^{12} + 54168 x^{11} - 126883 x^{10} - 165800 x^{9} + 611032 x^{8} + 53790 x^{7} - 1264603 x^{6} + 612252 x^{5} + 830632 x^{4} - 718892 x^{3} + 140512 x^{2} + 9184 x - 3136 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1503855547469708539369307904147456=2^{20}\cdot 3^{14}\cdot 7^{10}\cdot 101^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69.69$
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:  $2, 3, 7, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{2} a^{7} - \frac{1}{6} a^{6} + \frac{1}{6} a^{5} + \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{12} a^{11} - \frac{1}{6} a^{9} + \frac{1}{6} a^{8} + \frac{1}{6} a^{7} - \frac{1}{3} a^{5} - \frac{1}{6} a^{4} - \frac{5}{12} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{12} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{3}$, $\frac{1}{24} a^{13} - \frac{1}{24} a^{12} - \frac{1}{12} a^{10} + \frac{1}{12} a^{9} - \frac{1}{6} a^{8} - \frac{1}{4} a^{7} - \frac{1}{6} a^{6} + \frac{7}{24} a^{5} + \frac{7}{24} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{24} a^{14} - \frac{1}{24} a^{12} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{8} a^{6} + \frac{1}{4} a^{5} - \frac{3}{8} a^{4} - \frac{1}{4} a^{3} + \frac{1}{6} a^{2} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{48} a^{15} - \frac{1}{48} a^{13} + \frac{1}{24} a^{10} - \frac{1}{24} a^{9} + \frac{5}{24} a^{8} + \frac{5}{16} a^{7} - \frac{1}{24} a^{6} - \frac{13}{48} a^{5} - \frac{11}{24} a^{4} - \frac{1}{6} a^{3} + \frac{5}{12} a^{2} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{144} a^{16} - \frac{1}{144} a^{15} - \frac{1}{144} a^{14} - \frac{1}{144} a^{13} - \frac{1}{24} a^{12} + \frac{1}{72} a^{11} - \frac{1}{9} a^{9} + \frac{13}{144} a^{8} + \frac{67}{144} a^{7} - \frac{1}{48} a^{6} + \frac{49}{144} a^{5} - \frac{1}{6} a^{4} - \frac{17}{36} a^{3} + \frac{17}{36} a^{2} - \frac{2}{9} a + \frac{1}{9}$, $\frac{1}{128320562564887183491249648} a^{17} - \frac{27642277183576248405167}{64160281282443591745624824} a^{16} - \frac{45189136778579205573307}{128320562564887183491249648} a^{15} + \frac{14447694151524318170657}{1145719308615064138314729} a^{14} + \frac{196994849480150254861141}{21386760427481197248541608} a^{13} + \frac{1178656853625654965098967}{32080140641221795872812412} a^{12} - \frac{725403679750672182557035}{21386760427481197248541608} a^{11} - \frac{712802930389762146584327}{64160281282443591745624824} a^{10} + \frac{12289406405166724421195971}{128320562564887183491249648} a^{9} - \frac{7088719380865423267231667}{32080140641221795872812412} a^{8} - \frac{11228338083559160597240729}{42773520854962394497083216} a^{7} - \frac{25391273136951284976771763}{64160281282443591745624824} a^{6} - \frac{1935963575882742452602137}{7128920142493732416180536} a^{5} + \frac{31076256545847974525710697}{64160281282443591745624824} a^{4} - \frac{690372176158772939872888}{8020035160305448968203103} a^{3} - \frac{7732342679050582175718925}{16040070320610897936406206} a^{2} - \frac{780062910011092432985035}{16040070320610897936406206} a - \frac{108785669977728751366160}{381906436205021379438243}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $17$
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:  \( 970263885439 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_3:S_4$ (as 18T154):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 432
The 20 conjugacy class representatives for $C_3:S_3:S_4$
Character table for $C_3:S_3:S_4$

Intermediate fields

3.3.404.1, 6.6.7997584.1, 9.9.5539939488857088.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.8.3$x^{6} + 2 x^{3} + 6$$6$$1$$8$$D_{6}$$[2]_{3}^{2}$
2.6.8.3$x^{6} + 2 x^{3} + 6$$6$$1$$8$$D_{6}$$[2]_{3}^{2}$
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.12.12.23$x^{12} + 21 x^{11} + 21 x^{10} + 63 x^{9} + 36 x^{8} + 54 x^{7} + 90 x^{6} + 81 x^{3} - 81$$3$$4$$12$$S_3 \times C_4$$[3/2]_{2}^{4}$
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.12.10.6$x^{12} - 217 x^{6} + 11907$$6$$2$$10$$C_{12}$$[\ ]_{6}^{2}$
$101$$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$