↑

Properties

Label 18.0.62181896314...2704.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 19^{15}$
Root discriminant $18.46$
Ramified primes $2, 19$
Class number $1$
Class group Trivial
Galois group $S_3 \times C_3$ (as 18T3)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![83, -173, 348, 5, -526, 692, 46, -1329, 1007, 567, -970, 223, 260, -181, 13, 29, -11, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 11*x^16 + 29*x^15 + 13*x^14 - 181*x^13 + 260*x^12 + 223*x^11 - 970*x^10 + 567*x^9 + 1007*x^8 - 1329*x^7 + 46*x^6 + 692*x^5 - 526*x^4 + 5*x^3 + 348*x^2 - 173*x + 83)
 
gp: K = bnfinit(x^18 - x^17 - 11*x^16 + 29*x^15 + 13*x^14 - 181*x^13 + 260*x^12 + 223*x^11 - 970*x^10 + 567*x^9 + 1007*x^8 - 1329*x^7 + 46*x^6 + 692*x^5 - 526*x^4 + 5*x^3 + 348*x^2 - 173*x + 83, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} - 11 x^{16} + 29 x^{15} + 13 x^{14} - 181 x^{13} + 260 x^{12} + 223 x^{11} - 970 x^{10} + 567 x^{9} + 1007 x^{8} - 1329 x^{7} + 46 x^{6} + 692 x^{5} - 526 x^{4} + 5 x^{3} + 348 x^{2} - 173 x + 83 \)

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:  \(-62181896314367173832704=-\,2^{12}\cdot 19^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.46$
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:  $2, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{197} a^{15} - \frac{31}{197} a^{14} - \frac{44}{197} a^{13} - \frac{66}{197} a^{12} - \frac{89}{197} a^{11} - \frac{4}{197} a^{10} - \frac{86}{197} a^{9} - \frac{13}{197} a^{8} + \frac{66}{197} a^{7} + \frac{72}{197} a^{6} - \frac{22}{197} a^{5} + \frac{14}{197} a^{4} - \frac{50}{197} a^{3} - \frac{89}{197} a^{2} + \frac{41}{197} a - \frac{23}{197}$, $\frac{1}{16351} a^{16} + \frac{22}{16351} a^{15} - \frac{4445}{16351} a^{14} + \frac{1345}{16351} a^{13} + \frac{7445}{16351} a^{12} + \frac{7}{16351} a^{11} + \frac{1081}{16351} a^{10} + \frac{3703}{16351} a^{9} - \frac{5942}{16351} a^{8} - \frac{5492}{16351} a^{7} - \frac{3101}{16351} a^{6} + \frac{7713}{16351} a^{5} + \frac{4435}{16351} a^{4} + \frac{7505}{16351} a^{3} + \frac{5371}{16351} a^{2} + \frac{2347}{16351} a + \frac{85}{197}$, $\frac{1}{607126562864851721} a^{17} - \frac{3674735660822}{607126562864851721} a^{16} + \frac{771239323929358}{607126562864851721} a^{15} + \frac{40933238713491682}{607126562864851721} a^{14} - \frac{114591845695020095}{607126562864851721} a^{13} + \frac{169068378434964366}{607126562864851721} a^{12} + \frac{178465181992696086}{607126562864851721} a^{11} - \frac{123470617779064622}{607126562864851721} a^{10} + \frac{50133925688057394}{607126562864851721} a^{9} - \frac{132991319169520311}{607126562864851721} a^{8} - \frac{102337752542740727}{607126562864851721} a^{7} + \frac{106158676162560180}{607126562864851721} a^{6} - \frac{116600048836429019}{607126562864851721} a^{5} - \frac{50798510551928645}{607126562864851721} a^{4} - \frac{233131214303594375}{607126562864851721} a^{3} - \frac{199260239735998046}{607126562864851721} a^{2} + \frac{188873844535328897}{607126562864851721} a + \frac{728068576345995}{7314777865841587}$

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

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 6329.31955748 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-19}) \), 3.1.76.1 x3, 3.3.361.1, 6.0.109744.2, 6.0.2476099.1, 6.0.39617584.1 x2, 9.3.57207791296.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.39617584.1
Degree 9 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
2Data not computed
$19$19.6.5.5$x^{6} + 1216$$6$$1$$5$$C_6$$[\ ]_{6}$
19.6.5.5$x^{6} + 1216$$6$$1$$5$$C_6$$[\ ]_{6}$
19.6.5.5$x^{6} + 1216$$6$$1$$5$$C_6$$[\ ]_{6}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.19.2t1.1c1$1$ $ 19 $ $x^{2} - x + 5$ $C_2$ (as 2T1) $1$ $-1$
* 1.19.3t1.1c1$1$ $ 19 $ $x^{3} - x^{2} - 6 x + 7$ $C_3$ (as 3T1) $0$ $1$
* 1.19.6t1.1c1$1$ $ 19 $ $x^{6} - x^{5} + 2 x^{4} + 8 x^{3} - x^{2} - 5 x + 7$ $C_6$ (as 6T1) $0$ $-1$
* 1.19.6t1.1c2$1$ $ 19 $ $x^{6} - x^{5} + 2 x^{4} + 8 x^{3} - x^{2} - 5 x + 7$ $C_6$ (as 6T1) $0$ $-1$
* 1.19.3t1.1c2$1$ $ 19 $ $x^{3} - x^{2} - 6 x + 7$ $C_3$ (as 3T1) $0$ $1$
*2 2.2e2_19.3t2.1c1$2$ $ 2^{2} \cdot 19 $ $x^{3} - 2 x - 2$ $S_3$ (as 3T2) $1$ $0$
*2 2.2e2_19e2.6t5.2c1$2$ $ 2^{2} \cdot 19^{2}$ $x^{18} - x^{17} - 11 x^{16} + 29 x^{15} + 13 x^{14} - 181 x^{13} + 260 x^{12} + 223 x^{11} - 970 x^{10} + 567 x^{9} + 1007 x^{8} - 1329 x^{7} + 46 x^{6} + 692 x^{5} - 526 x^{4} + 5 x^{3} + 348 x^{2} - 173 x + 83$ $S_3 \times C_3$ (as 18T3) $0$ $0$
*2 2.2e2_19e2.6t5.2c2$2$ $ 2^{2} \cdot 19^{2}$ $x^{18} - x^{17} - 11 x^{16} + 29 x^{15} + 13 x^{14} - 181 x^{13} + 260 x^{12} + 223 x^{11} - 970 x^{10} + 567 x^{9} + 1007 x^{8} - 1329 x^{7} + 46 x^{6} + 692 x^{5} - 526 x^{4} + 5 x^{3} + 348 x^{2} - 173 x + 83$ $S_3 \times C_3$ (as 18T3) $0$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.